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7e0479bfefa6578c31a2dbeed0250ceb 18324d55648ea80bd9f245f3b9c40b28cf25e2de 33184 F20101129_AAABUP zheng_x_Page_070.pro 5a4f49f1667e77d13ca08684dbece422 4ab15c1bc8416a872cac2a605578fcc3aafd7340 SPACETIME PROCESSING IN MULTIANTENNA SYSTEMS By XIAYU ZHENG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 S2007 Xiayu Zheng To my parents and those who have helped me during this journey ACKNOWLEDGMENTS First and foremost, I thank my advisor, Dr. Jian Li, for her constant support, encouragement, and guidance. She is ah .1< willing to share her knowledge and life experience as an advisor, collaborator, and friend. Her enthusiasm and devotion to research have ahrl . inspired me along the way. I also thank Dr. Dapeng Wu, Dr. K~enneth C. Slatton, and Dr. David Wilson in the department of mathematics for serving on my supervisory committee and for their valuable comments. Their wonderful teaching has been beneficial to my research work. I owe special thanks to Dr. Petre Stoica at Uppsala University, Sweden, for his guidance in several interesting topics. I felt fortunate to have the opportunity to work with him and to benefit from his insightful ideas and constructive advice. My thanks go to Lingf Du, Bin Guo, Dr. Yi Jiangf, Zhipengf Liu, Dr. Guoqingf Liu, William Roherts, Dr. Yijun Sun, Dr. Yanwei W .ng_ Dr. Zhisong Wang, Ming Xue, Yao Xie, Dr. Hong Xiong, Xumin Zhu, and all other members in Spectral Analysis Lab for their help and friendship. This dissertation is dedicated to my parents, whose patience and unconditional love have been the strongest support to me during my whole life. TABLE OF CONTENTS page ACK(NOWLEDGMENTS .......... . .. .. 4 LIST OF TABLES ......... ... . 7 LIST OF FIGURES ......... .. . 8 LIST OF NOTATIONS ......... . 10 ABSTRACT ......... ..... . 12 CHAPTER 1 INTRODUCTION ......... ... .. 14 1.1 Spatial Filtering in MultiAntenna Systems .... .. .. .. 16 1.2 Spatial Diversity in MultiAntenna Systems .... ... .. 17 1.3 Spatial Multiplexing in MultiAntenna Systems .. .. .. .. 18 2 ADAPTIVE ARRAYS FOR BROADBAND COMMUNICATIONS IN THE PRESENCE OF UNKNOWN COCHANNEL INTERFERENCE .. .. .. 20 2.1 Introduction ......... . . 20 2.2 Problem Formulation ......... ... 22 2.2.1 System Model ......... . 22 2.2.2 Problem Formulation ......... .. 23 2.3 Adaptive Detection ......... .. .. .. 24 2.3.1 Adaptive Detection (AD) . ..... .. 24 2.3.2 Iterative Adaptive Detection ...... .... 31 2.4 Robust Adaptive Detection ......... .. 32 2.4.1 Exact Solution ......... ... 33 2.4.2 Approximate Solution ........ .. 35 2.5 Numerical Examples ......... . 36 2.6 Conclusions ......... .. . 39 3 MIMO TRANSMIT BEAMFORMING UNDER UNIFORM ELEMENTAL POWER CONSTRAINT .. ... . .. 49 3.1 Introduction ......... . .. .. 49 3.2 MIMO Transmit Beamforming ... .. .. .. 52 3.3 Transmit Beamformer Designs under Uniform Elemental Power Constraint 54 3.3.1 Problem Formulation and SDR ..... .. . 54 3.3.2 MISO Optimal Transmit Beamformer ... .. .. .. .. 56 3.3.3 The Cyclic Algorithm for MIMO Transmit Beamformer Design .. 57 3.4 FiniteRate Feedback for Transmit Beamformingf Designs .. .. .. .. 58 3.4.1 Scalar Quantization ........ ... .. 58 3.4.2 Adhoc Vector Quantization ... .. .. .. 60 3.4.3 Vector Quantization under Uniform Elemental Power Constraint 61 3.5 Average Degradation of the Receive SNR ... . .. 62 3.5.1 Maximum Average Receive SNR E{hw"2} .. .. .. 63 3.5.2 Approximate Value of E{v~vvs2 Vt E Si} . . . 64 3.5.3 Quantifying the Average Degradation of the Receive SNR .. .. 65 3.6 Numerical Examples ......... .. .. 67 3.7 Conclusion ......... ... .. 69 4 EFFICIENT CLOSEDLOOP SCHEMES FOR MIMOOFDM BASED WLANS 78 4.1 Introduction ........ ... .. 78 4.2 C'1I. .il., Model . ... . .. . 80 4.3 ClosedLoop MIMO WLAN System Design ... . .. 80 4.3.1 System Description ......... ... .. 80 4.3.2 Precoder and Equalizer Design ..... .... . 81 4.3.3 Successive Soft Decoding . .... .. 83 4.4 Precoder Quantization ........ ... .. 84 4.4.1 Scalar Quantization ........ ... .. 85 4.4.2 Vector Quantization . .... ... .. 86 4.5 Robust Transceiver Design in the TDD Mode ... ... .. 88 4.6 Numerical Examples ......... .. .. 89 4.7 Conclusions ......... ... .. 93 5 CONCLUSIONS AND FUTURE WORK .... ... . 98 APPENDIX A PROOFS FOR CHAPTER 2 ............ ....... 101 B PROOFS FOR CHAPTER 3 ........... ....... 102 C PROOFS FOR CHAPTER 4 ............ ....... 104 REFERENCES ......... . .. . 105 BIOGRAPHICAL SK(ETCH ......... .. .. 112 LIST OF TABLES Table page 21 Complexity comparison of the Viterbi equalizers .... .. .. 30 LIST OF FIGURES Figure page 11 Singleantenna system versus multiantenna systems: (a) SISO, (b) MISO, (c) SIMO, and (d) MIMO. .. ... . 15 12 Capacity of a 4 x 4 multiantenna system versus a single antenna system. .. 16 13 Basic diagram of receive spatial filtering for a narrowband multiantenna system. 17 14 Signal power for a 1 x 4 SIMO system versus a single antenna system. .. .. 18 21 BER vs. SNR without interference, for various block sizes: (a) NV=200, and (b) NV=60. ............ .......... ... 41 22 BER vs. SNR in the presence of 1 interference, for various block sizes: (a) NV=200, and (b)N1=60. ........... .......... 42 23 BER vs. SNR in the presence of 2 interference, for various block sizes: (a) NV=200, and (b)N1=60. ........... .......... 43 24 BER vs. SNR for various NV, and L, for various block sizes: (a) NV=200, and (b)N1=60. ............ ........... 44 25 BER vs. SNR in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, (b) NV = 60. ..... .. . 45 26 BER vs. SNR for various NV, and L in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. 46 27 BER vs. SNR for various NV, and L in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. 47 28 BER vs. SNR for variouS E in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. .. .. .. .. 48 31 Transmit power distribution across the index of the transmit antennas for a (4, 1) system. ......... .... . 53 32 Performance comparison of various transmit beamformer designs with perfect CSI at the transmitter: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. 71 33 Performance comparison of various transmit beamformer designs for the (8,8) MIMO case ........ ..... 72 34 Performance comparison of various transmit beamformer designs with 2bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. ... .. 73 35 Performance comparison of various transmit beamformer designs with 4bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. ... .. 74 :36 Performance comparison of various transmit heanifornier designs with 6bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. .. .. .. 75 :37 Performance comparison of various transmit heanifornier designs with 8bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. .. .. .. 76 :38 Performance comparison of various (2,1) MISO systems. ... .. .. 77 :39 Average degradation of the receive SNR for a (4,1) MISO system. .. .. .. 77 41 Transmitter design for MIMOOFDM hased WLAN. ... .. .. 82 42 Receiver design for MIMOOFDM hased WLAN. ... ... .. 82 43 Performance comparison of MIMO WLAN (108 Mbps) schemes for uncorrelated channels in the absence of quantization errors. .... ... .. 94 44 Output SNRs of the subchannels obtained via GMD, ITCD, and SVD, with input SNR=22 dB. ........ .... ........_ 94 45 Performance comparison of MIMO WLAN (108 Mbps) schemes for correlated channels in the absence of quantization errors. .... ... .. 95 46 Performance comparison of the proposed closedloop schemes for uncorrelated channels with scalar or vector quantization for GMD. ... .. . .. 95 47 Performance comparison of the proposed closedloop schemes for uncorrelated channels with scalar or vector quantization for ITCD. ... .. .. 96 48 Performance comparison of the proposed closedloop schemes for uncorrelated channels under channel nxisniatches in the TDD mode for GMD. .. .. .. .. 96 49 Performance comparison of the proposed closedloop schemes for uncorrelated channels under channel nxisniatches in the TDD mode for ITCD. .. .. .. .. 97 LIST OF NOTATIONS letters denote matrices, boldface lowercase letters denote vectors. Transpose and conjugate transpose (Hermitian) of matrix X (or vector x), respectively. Rank of X. Trace of X. Determinant of X. The matrix X Y is positive semidefinite. NVx NV Identity matrix. The vector or matrix with all elements being equal to The absolute value of a scalar x (real or complex). Twonorm of a vector or a matrix. Vectorization operator (stacking the columns of X on top of each other). The inner product between vectors x and y. The floor operation. A diagonal matrix whose diagonal is formed by the elements of x. The expectation operation. Belongf to. Exponential. Natural logarithm. Boldface uppercase X' (x'), X* (x*) rank(X) tr(X) X or det(X)  xl ,  X  vec(X) (x, y) = [ j Diag(x) x*y E [] E exp Inx nmin., f (r) arg nmin., f (r) arg nmax., f (r) R(X) RA~xN, CA~x" The nxininiun value of f (.) with respect to x.. The nmaxiniun value of f (.) with respect to x.. The optimal value of .r that achieve the nxininiun of function f(.r). The optimal value of .r that achieve the nmaxiniun of function f(.r). A subspace spanned by the columns of X. The set of M~ x NV matrices with real ar complexvalued entries, respectively. nd Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPACETIME PROCESSING IN MULTIANTENNA SYSTEMS By Xiayu Zheng August 2007 C'I I!r: Jian Li Major: Electrical and Computer Engineering Digital communications using multiple antennas have emerged and quickly developed as one of the breakthroughs in wireless communications. Multiantenna systems can provide spatial filtering, spatial diversity and spatial multiplexing, compared to a singleantenna system. We focus on solving some practical problems in the applications of multiantenna systems. First, we study the detection problems in singleinput multioutput (SIMO) broadband communication in the presence of unknown cochannel interference. We propose several adaptive beamforming methods, including adaptive detection (AD), robust adaptive detection (RAD) and their corresponding iterative versions. All of those methods are shown to not only outperform the conventional maximum likelihood sequence estimator (\!1.r510), but also have a lower computational complexity than the latter. Next, we consider multiinput multioutput (j\! [MO) transmit beamforming under uniform elemental power constraint. This is shown to be a hardtosolve nonconvex problem. We first solve this problem for the multiinput single output (jl\!lO) case and obtain a closedform optimal solution. Then we propose a cyclic algorithm for the MIMO case which uses the closedform MISO solution iteratively. The cyclic algorithm has a low computational complexity and is convergent very fast. Moreover, we address the problem of finiterate feedback to acquire the channel state information (CSI) at the transmitter via developing numerical and analytical quantization methods. Finally, we propose efficient closedloop physical 1.v;r schemes for high speed wireless local area networks (WLAN). By applying the recently introduced MIMO transceiver designs, i.e., the geometric mean decomposition (GMD) and the uniform channel decomposition (UCD), to the WLAN application, we obtain a much improved performance compared to the conventional singular value decomposition (SVD) based counterpart. Furthermore, a vector quantization method is proposed for finiterate feedback in explicit feedback mode and a robust structure is shown in timedivision duplex (TDD) mode. CHAPTER 1 INTRODUCTION During the past decade, wireless communications have seen a tremendous growth of research activities. Several factors have contributed to this phenomena. The immense success of wireless applications, including the cellular technology based wireless standards, e.g., global system for mobile communications (GSM) [1], IS95 Code Division Multiple Access (CDMA) [2]. The dramatic progress in verylargescale integration (VLSI) technology has made the implementations of sophisticated signal processing algorithms possible with low cost and low power. Under such circumstances, high data rate and high quality wireless communications, are of great interest with the increasing insatiate demands from the customers. The data rate of the next generation wireless local area network (WLAN) will be increased to hundreds of megabits per second [3], [4], or even 1 gigabit per second [5]. The bandwidth has become a scarce and expensive resource [5]. Consider a wireless system with a data rate of 1 gigabit per second. If we utilize a spectral efficiency of 4 h/s/Hz (16QAM [6]), a 250MHz bandwidth is needed. Such a large bandwidth can only be easily obtained over a high frequency range, for example, at 40GHz. The increased path shadowing at high frequency range has already made the wireless link unusable at that frequency [5]. The need for more spectrally efficient technology has become urgent. Digital communications using multiple antennas have emerged and quickly developed as one of the most significant breakthroughs in wireless communications, after the pioneering work by Winters [7], Telatar [8], Foschini [9], etc. Multiantenna systems are considered as a promising solution to the future wireless communications offering high data rate and high quality services. Compared to a singleantenna (singleinput singleoutput (SISO)) system, multiantenna systems have multiple antennas at the transmitter (multiinput singleoutput (jl\!lO)), at the receiver (singleinput multioutput (SIMO)), or at both sides (multiinput multioutput (jl\!IlO)), as shown in Figure 11. I :Tx ''~ Rx Tx[": hf Rx (a) (b) Tx Rx Tx Rx Nr Nt Nr (c) (d) Figure 11. Singleantenna system versus multiantenna systems: (a) SISO, (b) MISO, (c) SIMO, and (d) MIMO. Consider a multiantenna system with Not transmit and NV, receive antennas (Figure 11 (d)). In a frequencyflat fading channel, the channel H(t) can be modelled as a stationary and ergodic random process [10]. Without loss of generality, we remove the time index t, and the receive signal vector is written as: y = Hx + n, (11) where H E CN~x"t denotes the channel matrix, x E CNxtX is the vector of transmit signals, and n e CNe X1 is the noise component. We assume a Rayleigh fading channel, and n E CNi(0, a2 Nr) is the additive, circularly symmetric complex Gaussian noise with zero mean and covariance matrix a2 NT. The capacity of the multiantenna system is [8], [11] CanMo =,t(C)P ma Elog det, (Is +2 H~xH* bps/Hz (12) where Ex = E~xx*} is the covariance matrix of the transmitted signals, and Pt is the total transmit power. The capacity of singleantenna system with the same total transmit power is [11], [12] Csso=E ogdt + 2 ,bps/Hz (13) where the scalar h denotes the channel. We can see from Figure 12 that the 4 x 4 multiantenna system has a significant capacity improvement over a singleantenna system. 15 20 25 Figure 12. Capacity of a 4 x 4 multiantenna system versus a single antenna system. Based on applications, multiantenna systems can usually be categorized into the following classes. 1.1 Spatial Filtering in MultiAntenna Systems In cellular systems (e.g., GSM [1], IS95 CDMA [2], Enhanced Data rates for Global Evolution (EDGE)), cochannel interference arises due to the frequency reuse in the wireless channel, which will significantly reduce the user capacity of the system. Multiantenna systems can provide spatial filtering (the socalled beamforming). Spatial filteringf aims at enhancing the signal of interest and suppressing cochannel interference based on the differences in their spatial signatures (spatial locations) [13], [14], [15], and thus allowing ..a oressive frequency reuse to increase the user capacity. Figure 13 shows the basic diagram of a receive spatial filtering scheme for a narrowband multiantenna system, where the weight vector w =I ]T i2 IU I N S dutd acri: ng11 to the channel state information (CSI). For a broadband multiantenna system, the channel becomes frequencyselective and we may have to design a spacetime weight matrix. Spatial filteringf requires the CSI knowledge of the desired signal of interest and the 5 10 SNR (dB) y (t) y (t) Weight Adjustment Block Figure 13. Basic diagram of receive spatial filteringf for a narrowhand multiantenna system. cochannel interference. The training sequence is usually designed and transmitted before data sequence transmission to obtain the CSI knowledge [16], [17], [18], [19]. 1.2 Spatial Diversity in MultiAntenna Systems Due to the constructive or destructive effects of signals travelling through the multiple paths from the transmitter to the receiver, the signal power fluctuates in a wireless link, which results in channel fading [20]. C'!I. .1.11 I fading is traditionally considered as a pitfall of wireless transmission. Multiantenna systems, however, can provide spatial diversity, which can turn the channel fading into a benefit. The essence of spatial diversity techniques is to collect signals which fade spatially independently, making sure that a reliable communication is possible as long as one of the paths is strong. We can see from Figure 14 that the signal power of the multiantenna system is more stable and much stronger than that of the single antenna system. With multiple antennas at either the receiver or the transmitter, spatial diversity can he divided into two forms: receive diversity (for SIMO channel) and transmit diversity (for MISO channel) [6], [20]. In a MIMO system, both receive and transmit diversities are S15 ti 20 25 0_ 1x4 SIMO system using spatial diversity 1x1 SISO system 0 20 40 60 80 100 Channel Sample in Time Figure 14. Signal power for a 1 x 4 SIMO system versus a single antenna system. attainable. To measure the spatial diversity, an important performance metric is defined as [21] [20] [22]: Definition 1. Let P,(p) denote the average error r, p~l..1sl.:7:;i of a scheme est :ll..altonoise natio (SNR) p. the 7. e ;i;, gain of the scheme is .lo g P, ( p) d = him (14) ptoo log p For a MIMO system with Nt transmit and NV, receive antennas, the maximum achievable spatial diversity gain is d = Nt N, if the channels fade independently and the transmit signals are constructed suitably [2:3], [24], [25], [26], [27], [28], [29]. When the knowledge of the CSI is available at the transmitter and/or the receiver, array gain can usually be made available together with spatial diversity gain through coherent combining at the transmitter and/or the receiver, which will result in an increase of receive SNR. 1.3 Spatial Multiplexing in MultiAntenna Systems Another advantage of the multiantenna systems is that they can provide spatial multiplexing using the degree of freedom provided by MIMO channels [9], [:30], [:31], [20]. Spatial multiplexingf allows the transmission of multiple different data streams via the multiple independent paths generated by the multiantenna systems. When the knowledge of CSI is only available at the receiver side, Bell Laboratories L lied I SpaceTime (BLAST) systems are considered as efficient architectures in practical implementations [9], [:30], [:31]. Various transceiver designs [:32], [:33], [:34], [:35] have recently been proposed to improve capacity and performance by exploiting the knowledge of the CSI at both the transmitter and receiver. To measure the spatial multiplexingf, we define the multiplexingf gain as [21], [20]: Definition 2. Let R(p) be the dtheira te of a scheme the .scheme i~s R(p) r = lim (15) ptoo log p For a MIMO system with Nt transmit and NV, receive antennas, the maximum achievable multiplexing gain is r = min{Nvt, ir1 CHAPTER 2 ADAPTIVE ARRAYS FOR BROADBAND COMMUNICATIONS IN THE PRESENCE OF UNKNOWN COCHANNEL INTERFERENCE 2.1 Introduction Broadband wireless communication is of significant interest for high data rate access systems. However, the capacity of such a system is usually severely limited by intersymbol interference (ISI) and cochannel interference (CCI). The ISI is caused by multipath time dispersion due to the broadband channel. The maximum likelihood sequence estimator (j11.r810) [36] implemented with the Viterbi algorithm [37], [6] is considered an optimum method to combat ISI under the white noise assumption. CCI, though, is still a problem. In cellular systems, e.g., in the enhanced data rates for global evolution (EDGE) systems, due to cell splitting and cell size reduction to increase frequency reuse, CCI from users in other cells becomes the bottleneck of the radio link performance [19]. To suppress CCI, various methods have been proposed assuming that information on the cochannel interference is available [16], [17]. The multiuser MLSE [16] jointly detects all signals including those from cochannels. Although a reduced state detection algorithm is proposed in [16], its complexity is still much higher than that of the conventional MLSE [36], especially when there are several cochannel interference. A suboptimum approach presented in [17] modifies the branch metric in MLSE by using the autocorrelation of cochannel interference plus noise. The complexity of this method is similar to that of MLSE. In practice, however, the channel information or the autocorrelation of cochannel interference is alrws difficult to obtain, and therefore availability of training data for cochannels is required. Even worse, the system performance is very sensitive to the estimation accuracy of the the interference channel. Other joint equalization and interference suppression methods have been proposed without relying on any cochannel information assumption [19], [18]. In [18], a twostage hybrid approach separates the CCI reduction and ISI equalization into two steps. In the first step, a spacetinle filter is designed to nmaxintize the sigfnaltointerferenceplusnoise ratio (SINR). The filtered received signals are then equalized by the Viterbi algorithm to combat ISI. For this method, the resulted state for the Viterbi algorithm is usually larger than that in the 1\LSE, and thus so is the complexity. The decision feedback equalizer (DFE) approach was used in [19]. The receiver performs a nlininiunt niansquare error CCI suppression, while leaving the mitigation of ISI for a subsequent reduced state Viterbi processor. This method can reduce the complexity of the Viterbi equalizer in the second step. However, these methods are based on using training data to obtain the weight vector of the spacetinle filter. The longer the training data, the better the performance of the spacetinle filter, but the lower the transmission efheciency. We propose herein a number of data adaptive heanifornlingf methods to mitigate CCI and for symbol sequence detection. Our methods are based on the nmaxiniun SINR criterion. Since no training data is really needed for the computation of the spacetinle heanifornier, our methods are efficient front a data transmission standpoint. In a frequencyselective block fading channel, the proposed adaptive detection (AD) algorithm first obtains a spacetinle heanifornier by using the received data. After suppressing the CCI, we use the Viterbi algorithm for symbol detection or possibly reduce the problem to single symbol detection by carefully choosing the length of the heanifornier, which makes a tradeoff between performance and complexity. Note that AD assumes the separability (or, lack of correlation) of the desired signals and the interference plus noise. This assumption does not hold exactly for small block sizes. To deal with this problem, AD can he modified by using an iterative scheme, referred to as the iterative AD (IAD). In IAD, we first estimate the interference plus noise signals, and then nmaxintize an estimated SINR directly to obtain the heanifornier. IAD can he used to further improve the performance of AD. In practice, imperfect channel estimates may have a detrimental effect on the performance of heanifornling methods, including AD, which motivated us to propose a robust AD (R AD) method. R AD nmaxintizes the estimated desired signal power under the constraint that the "true" channel is within an uncertainty region built around the estimated channel. Due to the significant complexity of the exact robust solution, we use an approximate method that has a much lower complexity. In doing so, we obtain a much more robust detector which has the same order of complexity as AD. RAD can also be modified by using an iterative scheme, with the resulting algorithm referred to as IRAD. We show numerically that these adaptive methods outperform the conventional MLSE significantly in the presence of CCI. Furthermore, our methods can have a much lower computational complexity compared with the conventional MLSE. We give the system model and formulate the problem of interest for a singlecarrier broadband communication system in Section 2.2. In Section 2.3, we first present AD, for the small block size case, IAD is then proposed to iteratively improve the performance. Note that AD basically assumes that the signal channel estimate is errorfree. In Section 2.4, we propose RAD based on maximizing the signal power for the imperfect channel estimation case. RAD can also be improved iteratively, which leads to IRAD. The numerical examples are given in Section 2.5 to demonstrate the efficiency of our methods. Finally, Section 2.6 contains our conclusions. 2.2 Problem Formulation 2.2.1 System Model Consider a singlecarrier singleinput multipleoutput (SIMO) broadband communication system. The data symbols are transmitted via a SIMO link with one transmit antenna and NV, receive antennas over a frequencyselective block fading channel. During each data block, we transmrrit a. sequence of complex symbhols {so(ub)}if~t, whichh ar~e composed of Le preambhle symbnols {so(ul)}o _L,,, anld Le postamb~le symblols {so(b)} l. Th'le preamble and postamble symbols, which are known to the receiver, can be used to facilitate equalization of the desired signal channel [38]. To avoid interblock interference, we should have Lt '> L, where L is the channel order. In the presence of cochannel users, we assumr e that thle signal of interest (SOI) {so(ub)}~T4 + anld th~e cochlannrel signals s{Sk = t~l~+1, k = 1, 2, K, ar~e unlcorrelated with each oth~er. Thl~e cochlannel interference is unknown to the desired user and can be .Iinchronous or synchronous with respect to the SOI. The baseband model of the NV,x 1 received data vector, after baudrate sampling, can be described as L K L y(un) = holso(n ) + hkC 8 ~,( Ij Z z(n) l=0 k=1 l=0 ho(z )so(u) + e(u), (21) where zl is the unitdelay operator, ho(zl) = hoo + holzl + ... + hoLzL is the finite impulse response (FIR) vector transfer function of the desired channel, with hol hot1i 60, **OlN I = 0,1,...,L, and {e(ub)} _L t+1 denote the c~ochannel interference plus noise vectors. According to the block fading assumption, the desired channel vectors { hot }1= o and the cochannel vectors { hk L=0, k~ = 1, 2, .. ., K, remain constant within each block, but they are generally timevarying from one block to another. 2.2.2 Problem Formulation The conventional MLSE detection [36] of the transmitted data can be expressed as N+L {S^o(u)},_ = arg mm i) oz sou2 (22) Iso(n)}" 1 l where {s~o(u))}" are the estimated data symbols and ho(zl) is the estimated FIR transfer function of the desired channel. The optimization problem in (22) is rather complicated but it can be simplified by using Viterbi algorithm [37], [6], which is statistically optimal under the white noise assumption. The computational complexity of the Viterbi algorithm applied to (22) is still rather high due to the vector filtering by ho(zl) required in (22). A simpler approximate MLSE detector, which will be termed as AMLD, consists of solving the following minimization problem [39], [40]: {s~o(u))}"=, = arg mso Re>"S a(u) Toso~(u) tisou(n 1) ilz(ylu)l) ,(23) where hi(z) = hio + hizz + ... + hizz and {7z~lL=o are the coefficients of a scalar filter y(zl) = To + 71zl +. .+ LzL defined by the equality h*(z)ho(zl~ ) 7tzI (24) l=L (note that we have 71 = Ti*). AMLD converts the vector Viterbi equalization into a scalar one, and has a lower complexity [39]. However, for the statistical performance's sake, in what follows we will focus on the use of MLSE. In general, however, the assumption underlying (22) or (23), namely that the term e(u) is temporally and spatially white does not hold. This is the case, for instance, when e(u) comprises unknown cochannel interference. In such a case, the performance of (22) or (23) can be far from optimum. Adaptive beamforming methods [13], [14] are widely used in array signal processing to suppress strong interference and jammers. In our communication problem, by using properly designed beamformers, we can suppress the CCI to improve detection performance, as explained in the following schemes. 2.3 Adaptive Detection We propose herein an adaptive spacetime beamformingfbased solution to the problem of detecting {so(u)}" ,, in the presence of CCI plus white noise. 2.3.1 Adaptive Detection (AD) We apply a spacetime beamformer to the received data, whose transfer function is given by: where NV, is the delay length (or order) of the beamformer. The beamformer output is given by: g*(z )y(u) = g*(z )ho(z )So(u) + g*(z )e(u). (26) Depending on the scenario, the order NV, of g(zl) can be chosen in various vwsi~ (see the following discussion for details). Generally, the larger NV,, the more degrees of freedom (DOF) the beamformer possesses, at the expense of increased computations. We also note that it appears to be no restriction to assume that the beamformer is causal, as in (25). Indeed, any noncausal beamformer (i.e., one that contains positive powers of z) can be obtained from g(zl) by multiplying (25) and (26) with zd for some d > 0. Without loss of generality, therefore, we let g(zl) be of the form in (25). Let f(z ) = fo + flzl + ... + ~fy, az ~"! a g*(z1)hoz ). (27) We can detect the symbol stream by solving the following leastsquares (LS) problem associated with (26) using the Viterbi algorithm: N+(N,+L) (s~o@n)} 7 = arg mmn g*(z )y(u) f(z )so@~2. (28) Thus, the design of spacetime beamformer in (26) and the symbol detection in (28) compose the whole process of our AD. Note that the noise term in (28) is not necessarily white, and therefore the LS metric in (28) is not optimal. However, if the beamformer is suitably designed (see below), the SINR for (26) and (28) will be much higher than the SINR for (22) or (23). This means that the detection in (28) may have a better performance than the MLSE in (22) or the AMLD in (23), even though AD does not estimate the CCI or the noise properties explicitly. Without constraining the problem unduly we can assume that the symbol sequence is white. Then the SINR for (26) is given by SINR =(29) E{g*(zl)e(u)2} E{g*(zl)y(u)2 0.2 2'1a where of is the average power of (so@n)} and f = i fo i IN+ (210) (211) (which is no restriction since multiplication of (26) by a constant does not change the SINR). Then maximizing the SINR in (29) is equivalent to minimizing E{ g* (zl)y (n) 2 b In finitesamples, therefore, we would like to design the beamformer such that N+Lt mmi g*(z ~l)y'), n=N,Lt S.t. g Zx1) 0 x1) __ ,1 (212) f 2 = 1. This is a quadratic optimization problem with linear equality constraints that can be solved in closed form, as shown below. Let T ggr 1 (213) Then the objective in (212) can be written as y (u) N+Lt n=N,Lt y ( NV,) (214) where y (u) y ( NV,) (215) Next we note that N, L k=0 l=0 N, L +N, CC g~i0(jk) k=0 j=k g*(z )ho(z ) N+j=0 ~ k=0 (216) Let f2 be set to a fixed value, such as f 2 = 1, g = gT gT " g*lig, N+Lt 11 C n= N,Lt y*(u) ... y*(n N,) . where has = 0, for j Sf [0, L]. It follows from (216) that the constraint in (212) can be rewritten as: hio 0 ... O go Io hi, '. ... O gl f; hi, t t .= ha (217) 0 '. h, 0 0 g f where Hi is a (NV, + L + 1) x Nr (NV, + 1) matrix, g is a NrIV N 1) x 1 vector and f is a (NV, +L+ 1) x vector. Combining (214) and (217) leads to the following reformulation of the beamformer design problem: mmn g*Rg, s.t. Hig = f Ig)  f  2 = 1. (218) For Nr (NV, + 1) > NV, + L + 1 and under the mild condition that the elements of ho(z ) have no common zeros [41], the rank of Ho is (NV, + L + 1). Then, for fixed f, the solution to (218) can be readily shown to be [14]: g = 11 Ho(H~iR HoI) f (219) We note that the number of DOF (i.e., the number of free elements) in the beamformer vector g is Nr (NV, + 1). After the satisfaction of the linear constraints in (218), the number of DOF left is equal to DOFSINR = (1r 1) NVg + 1) L. (220) These are the DOF that can be used to maximize the SINR, and their number increases linearly with NV, and NV,. For example, for Nr, = 2 and NV, = L, we have only DOFSINR 1; but for NV, = 10, DOFSINR = 9(Ng + 1) L, which is fairly close to the total DOF= 10(NV, + 1). The capability of the beamformer to maximize the SINR obviously increases with NV, and NV,, but so does its computational complexity. Since the parameter Nr, is usually limited by the hardware implementation, we can not change it very frequently. Generally, we can choose the parameter NV, according to the following rules: (rl) The stronger the interference signals (or the larger the number of cochannel interference K), the larger the beamformer length NV,. (r2) The larger the number of receive antennas NV,, the smaller the beamformer length The condition of the cochannel interference signals can be coarsely known according to a prior knowledge, e.g., the users at the cell edge are generally much easier to be interfered by the cochannel signals than the ones in the middle of the cell. Another important aspect, which was not discussed so far, is the choice of f. To address this choice we note the following facts: (a) From a computational standpoint, we would like f(zl) to have only a few nonzero elements, ideally just one. For instance, let us ;?i that only fk, / 0, which means that fk = 1 (due to f2 = 1). Then the detection in (28) becomes: {S^o(u))}" = arg mmn ) gr(z )y(u~) so(n k)2, (221) {so(n)}* I n= k+1 which can be easily solved. In particular, if the symbol stream is uncoded, then (221) decouples in NV very simple singlesymbol detection problems (i.e., no Viterbi algorithm is actually needed)! Other rationales, however, may require that several coefficients of f(zl) are different from zero see below. (b) In practice, R may be rather different from the true covariance matrix, unless NV is rather large (e.g., NV > N,(NV, + 1)). In particular, if NV is only slightly larger than Nr (NV, + 1), then R might be rather illconditioned [42], which leads to a beamformer with a large norm g2. A large g2 may have a detrimental effect on the actual SINR. Let us assume that the vector f has L + 1 nonzero elements. From a computational viewpoint, the Viterbi algorithm part of AD is simpler if the nonzero elements of f are consecutive (which leads to only M~ states in the Viterbi algorithm, where M~ is the number of distinct values in the digital modulation constellation), we thus assume that: fk / 0 k e [j, j + L] (2 =r 0 k [j, j + L] for some j E [0,N1,+L L] and some L E [0,N1,+L]. The polynomial f(z ) corresponding to (222) is given by: f(z' )= z (fo +fizl +... + fgz ) z f(z ), (223) and the associated detection (see (28)) is: N+j+L {So@ ()} _ = arg mmn g*(z )y(u) f (z )so(n )2 (224) According to the discussion in point (b) above, we recommend choosing f so as to mnimizeu~ g2 St. 27 = ),Wheref = j fo ] f To describe how this can be done, let Bj be the matrix made from the columns j + 1 through j + L + 1 of R1Ho(H*R1Ho)1, that is: 0 }j Bj = RC RoHu gR~i o) I }L + 1 (225) L+1 Then g2 COTTOSponding to the choice of f(zl) in (223) can be written as: g2 = Bjf2. (226) Operation Number of Operations MLSE AD Metric Calculations M L+)M 1 Storage MemoriesMM ComparisonsMM Table 21. Complexity comparison of the Viterbi equalizers For fixed j, the vector f that minimizes (226), f = arg mi f*(B *Bj)f, (227) is given byr the minimum eigennvector of RBj and the corresponding minimum value of (226) is   g   2 in [BJB ], where Xmin [ den~otes the minimum? eigenvalue of []. Then we obtain j as: y = arg min Xmin[B *Bj]. (228) jE[o,1v,+LL] The proposed adaptive broadband beamformer is obtained by using (227) and (228) in (219) (that is, g = Bjf), and the corresponding detector of AD, is then given by (224). The computational complexity of AD can be measured on the basis of two 1 in r~ parts. Firstly, the computation of the beamformer. The main task here is the computation of Rl and of (l^IZ1H0)1, which has a complexity on the order O((NV, + 1)3 V3) and, respectively, O((NV, + L + 1)3) flops for each data block. Secondly, the Viterbi algorithm for symbol detection. The complexities of detection for each symbol in AD and in MLSE are compared in Table 1. Note that the metric calculation for each block in AD needs O(2NV(N, + 1)NV, + NV(L + 1)ME +1)) flops (see (224)), while in MLSE it needs O(NNI,(L + 1)ME 3+1) flops (see (22)). When I < L, AD can be made simpler computationally than AMLD or MLSE (compare (224) with (22) or with (23)) by properly choosing NV, and NV,, despite the need for the beamformer computation in AD. 2.3.2 Iterative Adaptive Detection Note that AD assumes that the desired signals and the interference plus noise are separable (or uncorrelated) after the beamforming to obtain (29). As we have discussed in the previous subsection (point (b)), when NV is only slightly larger than NV,(NV, + 1), the performance of AD can be severely degraded. The main reason is that the desired signals and the interference plus noise are no longer :: II l~y" separable in this case. We propose herein an iterative detection method, called iterative AD (IAD), which can make AD work well even in the small block size cases. Consider the beamformer design in (212) or (218). We minimize the objective function in the design problem based on the received data, which is only an approximate solution to the maximization problem of (29) under small block size conditions. Our IAD, however, solves the maximum SINR problem in (29) more directly based on the estimated interference plus noise samples. Enhanced estimates of the interference plus noise samples can be obtained by iterative detection. The IAD can be summarized as follows: (1) Initialization: Use AD to get an initial estimate of so(u), a = 1, 2, ... N, as b~o(u), a = 1, 2, ..., N. Set the iteration number to i = 0. (2) Iteration: Using the initial estimate &o (u), a = 1, 2, ... N, estimate the interference plus noise samples in one block as e~(u) = y(u) ho(z )s&o(u), n = Lt +1i,L + 2,...,NV+ Lt. (229) Then obtain the sample covariance matrix of the interference plus noise as N+Lt n=N,Lt *IU 20 Replacing R in (225), (227), (228) and (219) by Qe in (230), compute a new beamformer vector ge as ge = Q~e Ho(HiQe Ho) f ,, (231) and increase the iteration number: i = i + 1. (3) Symbol Detection: We obtain the data sequence estimate so(u), n= 1, 2,...,NV, by N+j+L (So ()} 7, = arg mmn) g (z ')y(u) f (z )so@ j) 2. (232) If i < NVite,, where NViter is the prescribed total iteration number, then we set s~o(u) = so(u), n = 1, 2, ..., N, and go to Step (2). We remark here that the length NV, can be different between the beamformer in the Initialization Step and that in the Iteration Step. The computational complexity of IAD increases linearly with NViter. More precisely, the computationally complexity of IAD is approximately (NViter + 1) times that of AD. 2.4 Robust Adaptive Detection The AD beamformer design, like the MLSE or AMLD, implicitly assumes that the channel estimate of ho(zl) is "perfect", that is, ho(zl) = ho(zl). This assumption will of course be violated in any practical communication systems (see, e.g., [43], [44], [45]). Nevertheless, AD as well as MLSE (or AMLD) can still be used, but the problem is that their performance might be rather sensitive to errors in ho(zl) (as the derivation of none of these methods take the fact that ho(zl) is different from ho(zl) into account). In this section, we propose a robust adaptive detection (RAD) method that makes the AD less sensitive to channel estimation errors. Let R be the theoretical covariance matrix corresponding to (215), under the assumption that the SOI and the cochannel interference signals are uncorrelated with one another, R is given by R = i E .y*() ... y*(n NV,) y ( NV,) = ofHoHi + Qe, (233) where of2 is the signal power, Qe is the covariance matrix of the interference plus noise, and Ho is defined similarly to the Ho in (217). In the finite sample case, the first step of our robust approach, inspired by [46], [47], is to reestimate ho by maximizing the estimated signal power, &j under the following natural constraints: max& s.t. It &2 Hoi {& ,~ho} ho ho 12 I E, (234) denote the reestimated channel vector and the corresponding matrix, and E is a user defined parameter. The maximization problem in (234) is equivalent to the following maximization problem (see the Appendix A): max Xmin[(H R 1Ho) 1], s.t. ho ho2 I E. (235) {ho} 2.4.1 Exact Solution The maximization problem in (235) is equivalent to mmn Amax[H~R Ho], s.t.  ho ha  2 I E, (236) {ho} where Amax [] denotes the maximum eigenvalue. A simple algebraic manipulation can be used to reformulate the equation (236) as a SemiDefinite Program (SDP) [48]: min c0, s.t. a~ > Amax [HER Ho], {0, ho} E > I ho h~O 2 23 7 which is equivalent to min a~, s.t. alI H~ll Ho > 0, {0, ho } E > Iho ho 12, (238) and hence further equivalent to min c0, s.t. lH >0 {0, ho} 0o E (ho ho)* >o 29 (ho o I We can solve the SDP in (239) using publicdomain software like SeDuMi [49]. The sonlution has a comnplexity on the ordePr of O/AgD~ Dy) fops [50], [51], where g O(i 1 De steitrto ubeD ,(NV, + 1) is the dimension of the unknown variables, D1 = (NV, + L + 1) + Nr (NV, + 1) and D2 r 1+I g(V + 1) are the dimensions of the two constraint matrices. Once the solution ho to (239) is obtained, we use it in the AD algorithm instead of ho, and the resulting algorithm is what we called RAD. The computational complexity of RAD can be much higher than that of AD. Consequently, we propose below an approximate solution to (235), which has a much lower complexity. 2.4.2 Approximate Solution Note that Xmin [(HiiR Ho) ]> (240) tr [HER1Ho] We propose to maximize the lower bound in (240) instead of the objective function in (235): mmn tr[HgR Ho], s.t. ho holl2 I E. (241) {ho} Note that tr [H Ri Ho] "1 0 .. o A1 A1 ..o 0 0 ... O R R1 ... O 0 0 ... O 0 0 ... O 0 0 ... O 0 0 ... O +... h 0 0 ... Ri N, N, Shirho, (242) where R 1, i~j = 0,1,2,...,N7,, are~t iNr x NrV~ submaltr~i~es of 11, an~d r is the summration of the (NV, + L + 1) matrices with dimension Nr (L 1) x Nr (L + 1) shown in (242). Thus, the minimization problem in (241) can be rewritten as mmn hirho, s.t. ho holl2 I E (243) {ho} The solution to (243) can be easily obtained by the "RCB als..i s~I lIn~ in [46], [47]. Note that for the "RCB al.IsIll ~ I n~ the complexity is on the order O(NJ~(L + 1)3) flops, which is much less than what was required to solve (239). We reconinend this approximate method to obtain ho, due to its lower computational complexity. We use ho in Equations (227), (228) and (219) to obtain the "robust" heanifornier, and then perform the symbol detection using (224). An important issue here is the selection of the user parameter E. Generally, when making this choice, we can consider the following rules: (r:3) The larger the sample length NV, the smaller the user parameter E. (r4) The larger the number of receive antennas NV,, the larger the user parameter E. The effect of E on the detection performance will be discussed in Section 2.5 in our numerical study. In the small block size case, we can use an iterative R AD (IR AD) to improve the detection performance, similarly to the way in which we derive IAD from AD. We remark that in IR AD, we first obtain ho and initialize with R AD to obtain an initial Q, as in (230), and we then use AD for iteration. The Q, based beanifornier is robust against channel errors [1:3] and hence using AD in the iterations is sufficient. The computational complexity of R AD and IR AD can he on the same order as that of AD and IAD if we choose carefully the length of heanifornier NV,. Similarly, when L < L, R AD or IR AD can he computationally simpler than AMLD or MLSE. 2.5 Numerical Examples To demonstrate the performance of the previously proposed methods, we compare thent via MonteCarlo simulations with MLSE in the presence of cochannel interference. In the simulations, we use the exponentially decaying Rayleighfading channel model in [:38], [52] to generate the block fading broadband channels for both the desired channel and the interference channel. In the said channel model, the delay profiles {hkl~j ILo, k= 0, 1, .. ., K, j = 1, 2, .. ., 1,N are independent, zeronlean, complex Gaussian random variable with the following variances: E hO,2 * L/2I= .5 (244) The NV, x 1 desired channel and cochannel vectors are independently generated based on (244). Without loss of generality, we assume that the signal power E{sk 2} = 1, k= 0, 1,. ., K and each cochannel interference has an average signaltointerference ratio (SIR) of 0 dB, i.e., E( {ho 2 SIR 1,k=1 .(245) E( {hke 12 The order of the channel for both the SOI and the interference is equal to L = 2. The symbol constellation for both the desired signal and the interference is QPSK(, i.e., M~ = 4. The number of the receive antennas is NV, = 4, and Le = max{L, NV,}. First, we consider the performance of AD and IAD with perfect channel estimation, i.e., the true channel is assumed known. In Figures 2123, we show the biterrorrate (BER) versus the signaltonoise ratio (SNR) in the presence of different numbers of interference (K = 0, 1, 2) for various block sizes (NV = 60, 200) for AD, IAD and MLSE. For the computational simplicity in the detection part, we take the delay length of the effective channel f(zl) as L = 0. The delay lengths of beamformers are NV, = 2 in Figures 21 and 22 and NV, = 5 in Figure 23, which are chosen according to the general rules (rl) and (r2) in Section 2.3.1. When there is no interference, Figure 21 shows that MLSE is better than both AD and IAD as expected. However, for each data block, the beamformer computation in AD is on the order O(33 3) + O(53) flops, and only a single symbol detection is needed. The complexity of the metric calculation in the MLSE is on the order O(4 x 3 x 43NV) for each data block, and it needs more memory storage and comparison operations (see Table 1). Hence, the complexity of MLSE is much higher than that of AD. When there are interference, we can see from Figures 22 and 23 that both AD and IAD outperform MLSE significantly. Although the heanifornler computation is on the order of O(6 4 ) + O(8 ) for Figure 23, the complexity of MLSE is still much higher than that of AD since MLSE requires more nienory storage and comparisons. Regarding the comparison of AD and IAD, AD is more sensitive to the small block size problem than IAD, especially when the cochannel interference are stronger. As expected, the proposed methods are most effective when there exist unknown cochannel interference. In what follows, we focus on the case of one dominant interference. Note that in Figure 22, when we increase the iteration number, little intprovenient can he achieved if NViter > 1. Hence we will choose NViter = 1 in what follows. With NViter = 1, the complexity of IAD is only twice that of AD, and thus still much lower than that of the MLSE. Figure 24 shows the influence of the heanifornier length NV, and of the effective channel da 1 0 length L on the BER performance of AD and IAD. When the block size is large (NV = 200), the performance of both AD and IAD improves as NV,increases. If the block size is small (NV = 60), on the other hand, the performance of AD degrades even when NV, is increased front 2 to 3. (The reason might he that the desired signal and the colored noise in (29) are not necessarily separable in the small block size case.) Concerning IAD, as it nmaxintizes the SINR hased directly on an approximate interference plus noise sample covariance matrix, its performance increases with the heanifornier length NV, even for NV = 60. Note that although the performance of both AD and IAD for L = 1 is usually slightly better than that for L = 0 with the same NV,, the computational complexity is approximately M~ times higher than that for L = 0. Second, we investigate the performance of our proposed methods under the more practical assumption of imperfect channel estimation. We obtain the imperfect channel estimates by adding a perturbation to the true channel: ha = ha + 0. 1 h,, (246) where h, is independently generated using the channel model in (244). For RAD and IRAD, the user uncertainty parameter is chosen as E = 6 ho$ (247) Figure 25 is obtained using 6 = 0.01, NV, = 2, and L = 0. From this figure, we see that all proposed methods are much better than MLSE even for imperfect channel estimation. Compared with Figure 22, AD degrades more than 5 dB in SNR, however, RAD improves the performance of AD significantly, resulting in a much smaller SNR loss compared to the perfect channel case. The iterative methods (IAD or IRAD) are robust against channel estimation errors and they perform quite well. In Figure 26 we show the BER performance of AD and of IAD with various NV, and L in the presence of channel estimation errors. It appears that choosing NV, = 2 and L = 0 (NViter = 1) gives the best overall compromise between performance and computational complexity. (Larger NV, is needed if more cochannel interference are present as in Figure 23.) Figure 27 is similar but for RAD and IRAD. IRAD is more robust to imperfect channel estimation and to the small block size problem than RAD, and even IAD compared to Figure 26 (b). Finally, we investigate the performance of RAD and IRAD for various values of the uncertainty parameter E (or, more precisely, 6 in (247)) in the presence of channel estimation errors. Figure 28 is obtained with NV, = 2 and L = 0 (NViter = 1). From these figures, we see that the effect of E (or 5) on the performance of RAD and IRAD is quite small. 2.6 Conclusions We have presented several adaptive beamforming methods based on the maximizing the SINR, for data detection in broadband communications in the presence of CCI. All proposed methods outperform the conventional MLSE significantly. Specifically, by carefully choosing the delay length of the beamformer, AD can achieve a good tradeoff between performance and computational complexity, and can even be implemented via single symbol detection, with a complexity much lower than that of the conventional MLSE. By iterating AD, we obtained IAD which can be used to improve the BER performance of AD at the cost of higher computational complexity. A robust AD, i.e., RAD was proposed to mitigate the problem induced by channel estimation errors. An iterative version of RAD, i.e., IRAD was also discussed. We believe that the excellent BER performance and the low computational complexity of our adaptive methods make them attractive detection methods for broadband communication systems. 10 e MLSE * AD 104~ .c IAD (iter=1) I:~ ~: < IAD (iter=2 *+ IAD (iter=3) 10s 10 5 0 5 10 15 SNR (dB) (a) 100 11 102 3 10 * AD 104~ c IAD (iter=1) :. < IAD (iter=2) *+ IAD (iter=3) 10s 10 5 0 5 10 15 SNR (dB) (b) Figure 21. BER vs. SNR without interference, for various block sizes: (a) NV=200, and (b) NV=60. 10 102 13 e MILSE I 4, 104~  ci IAD (iter=1) <1 IAD (iter=2) 1 + IAD (iter=3) I: :': 10s 5 0 5 10 15 20 SNR (dB) (a) 100 10 . 12 ' 10 e MLSE . * AD 104 ci IAD (iter=1) <3 IAD (ter=2) + IAD (iter=3) 10s 5 0 5 10 15 20 SNR (dB) (b) Figure 22. BER vs. SNR in the presence of 1 interference, for various block sizes: (a) NV=200, and (b) NV=60. 10  M 10 ' e MILSE a IAD (iter=1) < IAD (iter=2) + IAD (iter=3) 104 5 0 5 10 15 20 SNR (dB) (a) 100 11 12 e MILSE 2,~~ *AD . l IAD (iter1) 4 IAD (iter2) *+ IAD (iter3) 10 5 0 5 10 15 20 SNR (dB) Figure 23. BER vs. SNR in the presence of 2 interference, for various block sizes: (a) NV=200, and (b) NV=60. 10  10 ~ 104 : : e AD (Ng=2, Lbar=0) i s  AD (Ng=2, Lbar M( 10 5 4AD (Ng=3, Lbar=0) I : .e IAD (Ng=2, Lbar=0) 16 . IAD (N =2, L a=1) ..4. IAD (Ng=3, Lbar=0) t 10 5 0 5 10 15 20 SNR (dB) (a) 100 10  a1 AD (Ng=2, Lbar=, i 4AD (Ng=3, Lbar=0) .a ~ IAD (N =2, L a=0) ..a. IAD (Ng=2, Lbar~l ~ ..4 IAD (Ng=3, Lbar=0) 10s 5 0 5 10 15 20 SNR (dB) Figure 24. BER vs. SNR for various NV, and L, for various block sizes: (a) NV=200, and (b) NV=60. PM 10 2 W 10 104 5 0 5 10 15 20 SNR (dB) (b) Figure 25. BER vs. SNR in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, (b) NV= 60. SNR (dB) (a) 102 e AD (Ng=2, Lbar=0)n m AD (N8=2, Lbarl 4 AD (N = 3, L = 0) 10 4 IAD (Ng=2, Lbar=0) : ..a. IAD (Ng=2, Lbarl j ..4 IAD (Ng=3, Lbar= 0)I: : : 104 5 0 5 10 15 20 SNR (dB) (a) 100 11 eAD (Ng=,La=) a +021 AD (No=,Lbr0 . IAD(Ng=2, Lbar=0) ..a. IAD (Ng=2, Lbar1 ;, ..4 IAD (Ng=3, Lbar=0)\ W 10 5 0 5 10 15 20 SNR (dB) (b) Figure 26. BER vs. SNR for various NV, and L in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. 10  102 103 RAD (No=,Lbr0 . RAD (Ng=2, Lbar1 I i i ,i tRAD (No=,Lbr0 04 .4. IRAD (Ng=2, Lbar=0) ~k~` 10 .. IRAD (Ng=2, Lbar=l 4 IRAD (Ng=3, Lbar=0)I j j j j j j:j 10s 5 0 5 10 15 20 SNR (dB) (a) 100 10  12 10 +1 RAD (Ng=2, Lbar=0) .. RAD (N =2, L a=1) + RAD (Ng=3, Lbar=0) t o 14 .4. IRAD (NB=2, Lbar=0) 10 IRAD (N =2, L ,=1) 4. IRAD (Ng=3, Lbar=0) 10s 5 0 5 10 15 20 SNR (dB) (b) Figure 27. BER vs. SNR for various NV, and L in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. 10 ' W10 a RAD (60.05 +~ RAD (60.01 103 1 RAD (60.02) 1 +. **o IRAD (60.005) I ~II T*4  IRAD (G0.01) I: 1IRAD (60.02) 104 5 0 5 10 15 20 SNR (dB) (a) 100 11 10 : M 10 a RAD (6=0.005) + RAD (6=0.01) 10 S a RAD (6=0.02) * IRAD (6=0.005) :44 4 IRAD (G=0.01) + Sc IRAD (6=0.02) 104 5 0 5 10 15 20 SNR (dB) (b) Figure 28. BER vs. SNR for variouS E in the presence of channel estimation errors with 6 = 0.01, for various block sizes: (a) NV=200, and (b) NV=60. CHAPTER :3 MIMO TRANSMIT BEAMFORMING UNDER UNIFORM ELEMENTAL POWER CONSTRAINT 3.1 Introduction Exploiting multiinput multioutput (jl!IjlO) spatial diversity is a spectrally efficient way to combat channel fading in wireless communications. Although the theory and practice of receive diversity are well understood, transmit diversity has been attracting much attention only recently. Generally, the transmit diversity systems belong to two groups. In the first group, the channel state information (CSI) is available at the receiver, but not at the transmitter. Orthogonal spacetime block codes (OSTBC) [2:3], [24] have been introduced to achieve the maximum possible spatial diversity order. In the second group, the CSI is exploited at both the transmitter and the receiver via MIMO transmit heamforming, which has recently attracted the attention of the researchers and practitioners alike, due to its much better performance compared to OSTBC [20], [28]. Compared to OSTBC, MIMO transmit heamforming can achieve the same spatial diversity order, full data rate, as well as additional array gains. However, implementing MIMO transmit heamforming schemes in a practical communication system requires additional considerations. First, optimal transmit heamformers obtained by the conventional, i.e., the maximum ratio transmission (\l~rT) approach may require different elemental power allocations on the various transmit antennas, which is undesirable from the antenna amplifier design perspective. Especially in an orthogonal frequency division multiplexing (OFDM) system, this power imbalance can result in high peaktoaverage power ratio (PAPR), and hencewise reduce the amplifier efficiency significantly [5:3]. These practical problems have been considered in [5456] for new transmit heamfomer designs, and have also been addressed for transmitter designs in a downlink multiuser system [57]. Second, we need to consider how to acquire the CSI at the transmitter. Recent focus has been on the finiterate feedback techniques for the current conventional transmit beamforming [58], [59], [60], [61], [62], [63]. These techniques attempt to efficiently feed back the transmit beamformer (or the CSI) from the receiver to the transmitter via a finiterate feedback channel, which is assumed to be delay and error free, but bandwidthlimited. The problem is formulated as a vector quantization (VQ) problem [64], [65] and the goal is to design a common codebook, which is maintained at both the transmitter and the receiver. For frequenos lflat independently and identically distributed (i.i.d.) Raleigh fading channels, various codebook design criteria can be used and the theoretical performance (e.g., outage probability [60], operational ratedistortion [62], capacity loss [63]) can be analyzed for the multiinput singleoutput (jl\!lO) case. The feedback schemes can be readily extended to the frequencyselective fading channel case via OFDM. The relationship among the OFDM subcarriers can also be exploited to reduce the overhead of feedback by vector interpolation [66]. We address the aforementioned problems as follows. First, we consider MIMO transmit beamformer design under the uniform elemental power constraint. This is a nonconvex optimization problem, which is usually difficult to solve, and no globally optimal solution is guaranteed [54]. Generally, we can relax the original problem to a convex optimization problem via SemiDefinite Relaxation (SDR). The relaxed problem can be solved via public domain software [49]. We can then obtain a solution to the original nonconvex optimization problem from the solution to the relaxed one by, for example, a heuristic method [48] (referred to as the heuristic SDR solution). Interestingly, we find out that in the multiinput singleoutput (jl\!lO) case, the optimal solution has a closedform expression and is referred to as the closedform MISO transmit beamformer. (Similar results have appeared in [5456] for equal gain transmission (EGT).) We then propose a cyclic algorithm for the MIMO case which uses the closedform MISO optimal solution iteratively and the solution is referred to as the cyclic MIMO transmit beamformer. The cyclic algorithm has a low computational complexity and is shown via numerical examples to converge quickly from a good initial point. The numerical examples also show that the proposed transmit heamformingf approach outperforms the conventional one with peak power clipping. Meanwhile, the cyclic solution has a comparable performance to the heuristic SDR hased design and outperforms the latter when the rank of the channel matrix increases. Second, we consider finiterate feedback schemes for the proposed transmit heamformer designs. A simple scalar quantization (SQ) method is proposed; by taking advantage of the property of the uniform elemental power constraint, the number of parameters to be quantized can he reduced to less than one half of their conventional counterpart. VQ methods are also discussed. Although the existing codebooks [58], [59], [60], [62], [6:3] can he used with some modifications by the MISO closedform solution, the performance may not he optimal since they do not take into account the uniform elemental power constraint in the codebook construction. We propose in this chapter a VQ method for transmit heamformer designs whose codebook is constructed under the Uniform Elemental Power constraint (referred to as VQITEP). The generalized Lloyd algorithm [64] is adopted to construct the codebook. When the number of feedback bits is small, VQITEP performs similarly to the conventional VQ (CVQ) method without uniform elemental power constraint. For the MISO case, we further quantify the performance of VQITEP hy obtaining an approximate closedform expression for the average degradation of the receive signaltonoise ratio (SNR). It is shown that this approximate expression is quite tight and that we can use it as a guideline to determine the number of feedback bits needed in practice, for a desired average degradation of the receive SNR. The remainder of this chapter is organized as follows. Section :3.2 describes the conventional MIMO transmit heamforming and its limitations. Section :3.3 presents our closedform MISO and cyclic MIMO transmit heamformer designs under the uniform elemental power constraint. In Section :3.4, we consider the finiterate feedback schemes, where a simple SQ method and VQITEP are proposed. In Section :3.5, we focus on the MISO case and quantify the average degradation of the receive SNR caused by VQITEP hy obtaining an approximate closedform expression. Numerical examples are given in Section :3.6 to demonstrate the effectiveness of our designs. We conclude the chapter in Section :3.7. 3.2 MIMO Transmit Beamforming Consider an (Nt, NV,) MIMO communication system with Nt transmit and NV, receive antennas in a quasistatic frequency flat fading channel. At the transmitter, the complex dtaif symbold EC is mlodulatefdJ by. the beamflormerl we~ =11 ZIt t,2 ... 11L,Nt i; and then transmitted into a MIMO channel. At the receiver, after processing with the combining vector we ', iF .. 1 ,~N ]t he sampled combined brasebad signal is given by Y = wr [Hwts + n] (:31) where H E CNe x" is the channel matrix with its (i, j)th element hij denoting the fading coefficient between the jth transmit and ith receive antennas, and n E CNe xl is the noise vector with its entries being independent and identically distributed (i.i.d.) complex Gaussian random variables with zeromean and variance O.2. Note that in the presence of interference, i.e., when n is colored with a known covariance matrix Q, we can use prewhitening at the receiver to get Y = wr QHz n.(32 Hence (:32) is equivalent to (:31) except that H in (:31) is now replaced by QsH and the whitened noise has unit variance. Without loss of generality, we focus on (:31) hereafter. The transmit heamformer we and the receive combining vector we in (:31) are usually chosen to maximize the receive SNR. Without loss of generality, we assume that S0.3   0.2  0.15 0.05111 1 1.5 2 2.5 3 3.5 4 Index of Transmit Antennas Figure 31. Transmit power distribution across the index of the transmit antennas for a (4, 1) system. wt1 2 = 1, IWr 12 = 1, and E{s2} = 1. Then the receive SNR is expressed as p vrs vt~I~I* r* = (33) E= {w~x*n2 2 To maximize the receive SNR, the optimal transmit beamformer is chosen as the eigenvector corresponding to the largest eigenvalue of H*H [62] (referred to as MRT in [54]), which is also the right singular vector of H corresponding to its dominant singular value. The optimal combining vector is given by w, = which can be shown to be the left singular vector of H corresponding to its dominant singular value (referred to as maximum ratio combining ( \! RC) in [54]). Thus, the maximized receive SNR is p = mx(H*H), Where Xmax() den~otes the maximum? eigen~value of a m~a~trix. The cova~rianlce matrix of the transmitted signal is R = E {wess*w; } = wtwt*. (34) The average transmitted power for each antenna is Pi = Rei = Wt,i2, i = 1,2,... ,Nsv, (35) where Rii denotes the ith diagonal element of R. (Note that if the constellation of a is phase shift keying (PSK(), Pi represents the instantaneous power.) The average power Pi may vary widely across the transmit antennas, as illustrated in Figure 31, which shows a typical example of transmit power distribution across the antennas. The wide power variation poses a severe constraint on power amplifier designs. In practice, each antenna usually uses the same power amplifier, i.e., each antenna has the same power dynamic range and peak power, which means that the conventional MIMO transmit beamforming can suffer from severe performance degradations since it makes the power clipping of the transmitted signals inevitable. 3.3 Transmit Beamformer Designs under Uniform Elemental Power Constraint We consider below both MIMO and its degenerate MISO transmit beamformer designs under the uniform elemental power constraint. 3.3.1 Problem Formulation and SDR Given MRC at the receiver, maximizing the receive SNR p in (33) under the uniform elemental power constraint is equivalent to: max Hwt2, subject to  Wt,i2 i= 1, 2,...,NVt. (36) This is a nonconvex optimization problem, which is usually difficult to solve, and no globally optimal solution is guaranteed [48, 54, 67]. The problem in (36) can be reformulated as max tr(RG), fR) subject to Rii= =12..4 R E 0, rank(R) = 1, (37) where G n H*H E CNtx"t,R E CNexrv, and the inequality R > 0 means that the matrix R is positive semidefinite. Note that in (37), the objective function is linear in R, the constraints on the diagonal elements of R are also linear in R, and the positive semidefinite constraint on R is convex. However, the rankone constraint on R is nonconvex. The problem in (37) can be relaxed to a convex optimization problem via SemiDefinite Relaxation (SDR), which amounts to omitting the rankone constraint yielding the following SemiDefinite Program (SDP) [50]: max tr(RG), fR) subject to Ra= i ,,..& R > 0. (38) The dual form of (38) is given by [48] {x} subject to diag~x} G > 0, (39) where x E 7t~eX t c = l y with l y, denoting an Aidimensional all one column vector, and diag~x} is a diagonal matrix with x on its diagonal. The problem in (39) is also a SDP. Both (38) and (39) can be solved by using a public domain SDP solver [49]. The worst case complexity,, of,, solvng 39 is O(fsl'.) [51]. We can obtain the optimal solution to (39), whose dual is also the optimal solution to (38). Assume that the optimal solution to (38) is Ropt. Then tr {RoptG} > Hwt2 foT any we under the uniform elemental power constraint. If the rank of Ropt is one, then we obtain the optimal solution wy to (36) as the eigenvector corresponding to the nonzero eigenvalue of Ropt. If the rank of Ropt is greater than one, we can obtain a suboptimal solution wy from Ropt via a rank reduction method. For example, the heuristic method in [48] chooses w~ as the eigenvector corresponding to the dominant eigenvalue of Ropt* The Newtonlike algorithm presented in [68] uses the SDR solution as an initial solution and then uses the tangentandlift procedure to iteratively find the solution satisfying the rankone constraint. However, the approximate heuristic method is preferred, as shown in our later discussion, due to its simplicity. Interestingly, we show below that the optimal solution to (36) has a closedform expression for the MISO case. Moreover, we propose a cyclic algorithm for the MIMO case which uses the closedform MISO optimal solution iteratively. The cyclic method has a low complexity and numerical examples in Section 3.6 show that it converges quickly given a good initial point. Furthermore, we also show in Section 3.6 that the performance of the cyclic algorithm is comparable to that of the Heuristic SDR solution and in fact better when the rank of the channel matrix is large. Hence, the former is preferred over the latter in practice. 3.3.2 MISO Optimal Transmit Beamformer Let h E Clx"t be the row channel vector for the MISO case. We consider the maximization problem in (36) hwt2 = hwt2 Nvt 2 Nt 2Nt2 i= 1 i= 1 i= 1 where the equality holds when we = ej~h* j~ A w ej#, With _ej~h* denoting the unitnorm column vector having the angles of h*, and E [0, 2xr). Note that the optimal solution is not unique due to the angle ambiguity, yet we may take wy as the optimal solution to (36) for simplicity. (This result can also be found in [5456] for EGT.) 3.3.3 The Cyclic Algorithm for MIMO Transmit Beamformer Design The original maximization problem for (36) is max wf Hw 2, {Wr~wt} 1V ~ we= 1. (311) Inspired by the cyclic method (see, e.g., [69]), we solve the problem in (311) in a cyclic way for the MIMO case. The cyclic algorithm is summarized as follows: (1) Step 0: Set w, to an initial value (e.g., the left singular vector of H corresponding to its largest singular value). (2) Step 1: Obtain the beamformer wt that maximizes (311) for we fixed at its most recent value. By taking w~H as the "effective MISO channel," this problem is equivalent to (36) for the MISO case. The problem is solved in (310) and the optimal solution is: we = ejZLHlw,. (312) (3) Step 2: Determine the combining vector w, that maximizes (311) for we fixed at its most recent value. The optimal wr is the MRC and has the form: Hwt we (313) IHwt Iterate Steps 1 and 2 until a given stop criterion is satisfied. An important advantage of the above algorithm is that both Steps 1 and 2 have simple closedform optimal solutions. Also the cyclic algorithm is convergent under mild conditions [69]. We remark here that the cyclic algorithm is flexible and we can add more constraints on we or wt. A useful one is the uniform elemental power constraint on the receive anlten~na~s (or eqlual ga~in c~ombin~ing (EG C) [54, 59]),l i.e.,  w,7,1  = ,T i = 1, 2 Nr  Thenl we only have to m~odifyi (313) as w,, = ejZci' we in? Step 2 of each ite1ration?. Given a good initial value (e.g., the one as given in Step 0), the cyclic algorithm usually converges in a few iterations in our numerical examples, and the computational complexity of each iteration is very low, involving just (312) and (313). 3.4 FiniteRate Feedback for Transmit Beamforming Designs In the aforementioned transmit beamformer designs, we have assumed that the transmitter has perfect knowledge on the CSI. However, in many real systems, having the CSI known exactly at the transmitter is hardly possible. The channel information is usually provided by the receiver through a bandwidthlimited finiterate feedback channel, and SQ or VQ methods, which have been widely studied for source coding [64], [65], can be used to provide the feedback information. To focus on our problem, we assume herein that the receiver has perfect CSI, as usually done in the literatures [58, 59], [60], [62], [63]. 3.4.1 Scalar Quantization Note that the transmit beamformer we under the uniform elemental power constraint can be expressed as we(So,, ,01e1 (314) where the transmit beamformer wt(8o, , HNt1) is a function of NVt parameters {Oi, 8i E [0, 2xr) } o Via simple manipulations, we obtain 1 ey Wt 80, *, HNt1) CiBo SeeOwt(H1, HN l), (315) where Os = Os 8o, Hi E [0, 2xT), i = 1, 2, NVt 1. Since  Hwt(Bo, , ON1l) 2 Hwt(B1, , H#,1) 12, We can reduce one parameter and quantize wt(81, , HNt1) instead of wt(80o., H , u1) Denote 1 31 we(0"', 0 ) =J,1 (316) where H ,O , the number of quantization levels and feedback index of Os, respectively, and where Bi is the number of feedback bits for Os. After obtaining the transmit beamformer from (310) or the cyclic algorithm in Section 3.3.3, we quantize the parameters 04 to the " round off) grid points 8 ', i = 1, 2, .. ., NVt 1. Hence for this scalar quantization scheme, we need to send the index set (nl, n82., N,1) from the receiver to the transmitter, which requires B = E Bi bits. The receive combing vector is w, TIhe choice of {Be}~, is known as a counlting problem [70], which has" C"+fN'2 (+ 2) COmbinations. The optimal set {Bi)} is the one that maximizes Hwt (0" ,..., d4,_, )2. However, this exhaustive search is too complicated for practical applications. One simple suboptimal approach is to make Bi approximately equal. Specifically, let Bay = ,= + 1 and Ns, = B., (Nsi 1). Then we can let Bi = B,, bits for the first Ns parameters {84}, ,1 and Bi = B,, bits for the remaining (N~T 1)_ NS paramneters {Oi} t~. We remark here that for the conventional MIMO transmit beamformer without uniform elemental power constraint, the SQ requires about twice as many parameters. In this case, the transmit beamformer is expressed as w e(Ao, 8o, ..., ANt_1, HNt_1) = (317)1 7 where Ai, Ai E [0, 1] is the ith amplitude and Os, Os E [0, 2x) is the ith phase of the transmit beamformer vector, respectively, and hence there are totally 2NVt parameters. 3.4.2 Adhoc Vector Quantization Vector quantization can be adopted to further reduce the feedback overhead. In this case, both the transmitter and the receiver have to maintain a common codebook with a finite number of codewords. The codebook can be constructed based on several criteria. One approach is to directly apply the existing codebooks (e.g., [58, 59], [60], [62], [63]) constructed for the conventional transmit beamformer designs obtained without the uniform elemental power constraint. Among them, the criteria (e.g., [58], [62], [63]) that can be implemented by the generalized Lloyd algorithm can ahrlw lead to a monotonically convergent codebook. The generalized Lloyd algorithm is based on two conditions: the nearest neighborhood condition (NNC) and the centroid condition (CC) [6264]. NNC is to find the optimal partition region for a fixed codeword, while CC updates the optimal codeword for a fixed partition region. The monotonically convergent property is guaranteed due to obtaining an optimal solution for each condition. Maximizing the average receive SNR is a widely used criterion to design the codebook [58, 60, 62] and will also be adopted here for codebook construction. Some modifications are still needed as below when the uniform elemental power constraint is imposed. Let a codebook constructed for the conventional transmit heamformingf he W := {fyi, TV, WN, }, where Nz. = 2B is the number of codewords in the codebook W, and B is the number of feedback bits. The receiver first chooses the optimal codeword in the codebook as: y" = arg max Hfy2, 3 vE w where the operator arg max returns a global maximizer. Then we need to feed back the index of fy" from the receiver to the transmitter, which requires B hits. The transmit heamformer satisfying the uniform elemental power constraint is obtained as: wad = 6eL~O (39 nd. the receive combining vector is we =~ However; the codebooki W may not he optimal for our proposed transmit heamformer designs, since it is adhocly constructed without the uniform elemental power constraint (referred to as the adhoc vector quantization (AVQ) method). 3.4.3 Vector Quantization under Uniform Elemental Power Constraint Like AVQ. herein we also maximize the average receive SNR, while the codebook is constructed under the uniform elemental power constraint (referred to as "VQUEP"). For a given codebook W := { W ~, W2, ... N, }, the receiver first chooses the optimal transmit heamformer as: we~pt = arg max  Hw\  , (320) wNEW and the corresponding vector quantizer is denoted as wort = Q(H). Then we need to feedback the index of wort from the receiver to the transmitter with log2 1V, = B hits, and the rece:ivei comnbininlg vect:or is we = ii. Now the design problem becomes findings the codebook, which can he constructed offline as follows. First, we generate a training set {H1, H2, .. ,Hg} from a sufficiently large number NV of channel realizations. Next, starting from an initial codebook (e.g., a codebook obtained from the conventional transmit beamformer designs or one obtained via the splitting method [64]), we iteratively update the codebook according to the following two criteria until no further improvement is observed. (1) NNC: for given codewords {vivs} 7, assign a training element H, to the ith region Si = {H, : Haw 2 > Hnaw 2, Vj / i}, (321) where Si, i = 1, 2, ... N,, is the partition set for the ith codeword ws. (2) CC: for a given partition Si, the updated optimum codewords {v~vs} ,N satisfy weT = arg max Es [HnT;v 2Hn t SI] , subject to  m23_22) for i = 1, 2, ... N,. Let Ri = Es[1!"E adR/ be Hermitian square root of Ri. A simple reformulation results in wei = arg max IR w ,ll~ subject to  m23_23) TIhis problem is identical to (36) (H is replaced by R l/2) and canl be efficienltly solved by the cyclic algorithm proposed in Section 3.3.3. 3.5 Average Degradation of the Receive SNR For frequency flat i.i.d. MISO Rayleigh fading channels, various analysis approaches have been proposed to quantify the vector quantization effect (outage probability [60], operational ratedistortion [62], capacity loss [63], etc.). These analyses provide theoretical insights into the vector quantization methods and can serve as a guideline for determining the optimum number of feedback bits needed for the conventional transmit beamforming. We quantify below the effect of VQUEP with finitebit feedback on our closedform MISO trasmt bamorer1,,C, design. Let h, ~ (, ojf Iw,). Without loss of generality, we assume i = 1. The average degradation of the receive SNR is defined as: D, = E{hw,"2 hQ(h)2) = E{hw;2} (h E iA)E{hwsZ2h E &}, i= 1 SE{hw;2 PV Vt iCe F:)E{vfws2 V t 'j; )}E{h2}, (324) i= 1 where Si = {h : hws2 > hwy2,j 1 > S the partition set (or Voronoi cell) for the ith codeword v~v, iS = vL : vt = ~, he & },; P(h t iS) is the probability that a channel realization h belongs to the partition Si, and the last equality is due to the independence between h and the normalized vector h/h [62], [70]. Obviously, we have P(h a s)> = Pvt, E Si). 3.5.1 Ma xi~m um Aver\, age r: Tr Re ev N R E{hw 021 Using wy in (310), we get: E {hw,2} = ME{(hi +&2 Nt h12) =Var{Ihl}+NtE2 1}l, (325) where the last equlality is duec to the i.i.d. property oft {hs} k. The h  in (325) has the probability density function (pdf) as follows [71]: 2x x2 flh (x) = exp { 2 }, x > 0. (326) The mean and variance of hl are, respectively, E{hl}= (327) and Va h} a (328) Combining (327) and (328) into (325), we get 3.5.2 Aproimt Value:C of. E{v 4"i2 V t E Si} Note that the vector vt is considered as uniformly distributed on the unit hypersphere RN" [58, 60], [62], [63]. For a fixed codeword wei E R, the random variable yi = v*vys2 has a beta distribution Beta(1, NVt 1) [63], with the pdf: fr,(x) = (Not 1)(1 X)Nt2, O < x < 1. (330) Now we consider the conditional density frzl veg (X). Generally, each Voronoi cell [58, 60], [63], [64] obtained from the generalized Lloyd algorithm has a very complicated shape and it is difficult to obtain an exact closedform expression for f ,lvtE (X). We adopt herein the approximate method used in [60, 63] to analyze the problem at our hand. When NV, is reasonably large, we can approximate the probability P(vt E Si) as P(vt E Si) ~ , Vi. The Voronoi cells can be considered as identical to each other. We then approximate each Voronoi cell Si as a spherical segment on the surface of a unit hypersphere: where a =~~l~ + is the maximu average value of vf I* 2 ach~ieved by perfect feedback in our MISO transmit beamformer design, and the parameter 6 > 0 is the minimum value of vf#42 in each Voronoi cell. We need to solve the following equation related to B to obtain 5: Pvt, E Si) P(6 < yi < a) f (x)dx = 2 . (332) Using the pdf in (330), we get b 1 [2B + (1 a)"t ] "t (333) Thus, for the Voronoi cell Si, we approximate the conditional pdf of yi as fy, (x) ,6 < x < a, P (vt E Oi), /as ved,(x) (334) where 1, 6 < x < a, l~s~a(X>= 0, otherwise, is the indication fumetion. From the conditional pdfl f ~ivte,(x) in (334), we obtain (335) Jn^x 2B(Nt, 1)(1 X)Nt2d 1 + N 2U [(1 a)"~ (1 6)" ] . E{vt*firs2 Vt E Si (336) 3.5.3 Quantifying the Average Degradation of the Receive SNR Now we quantify the average degradation of the receive SNR in (324) using the approximate conditional pdf f~lvtess(x). From (336), we observe that the average receive SNR To is To(B) =) P(vt )E{E~v vlv42 VtE A}E{h2) i= 1 =1+ 2" [(1 )" (1 )Y.] Neo i= 1B[ t = Nef + Nt 1) 2 1 a1(2" (1 al)"l a o. (337) Combining (329) and (337) into (325), we obtain the following proposition: Proposition 3.5.1. For i.i.d. M~ISO Rarl /ple fading channels, the average degradation of the receive SNR, for an Neiantenna transmit beamforming system with an NV, = 2Bsize VQ UEP codebook, can be applrox~imated as: D,(B ~ No 1) 2 (2" +(1 a) ) t (1 a)Nt af Net(1 aUef. (338) The average degradation of the receive SNR in (338) can be proven to be monotonically decreasing with respect to nonnegative real number B (see Appendix B). Given a degradation amount Do, this proposition provides a guideline to determine the necessary number of feedback bits. That is, we can alrws find the optimum integer number of feedback bits B (via, e.g., the Newton's method) with the average degradation D,(B) of the receive SNR being less than or equal to Do. Similarly, the average receive SNR in (337) can be shown to be monotonically increasing with respect to B, and we can determine the needed number of feedback bits with the average receive SNR being less or equal to a desired yj. Although our analysis shares some similar features to those in [55, 56], our results are more accurate (see Section 3.6). In [55, 56], the authors found the pdf of (4 = 1  (hr~v42/hw;2) Via making more approximations. Under highresolution approximations, the average degradation of the receive SNR given in [55, 56] has the form: D,(B) ~ E{hwf2}E{(4} [ x Nt 1 2n Both (338) and (339) are compared with numerically determined average receive SNR loss at the end of the next section and (338) is shown to be more accurate than (339). 3.6 Numerical Examples We present below several numerical examples to demonstrate the performance of the proposed MISO and MIMO transmit beamformer designs under the uniform elemental power constraint. We assume a frequency flat Rayleigh channel model with E{hij2} = 1, i = 1, 2,..., Nr, j = 1, 2,..., Ns. In the simulations, we use QPSK( for the transmitted symbols. First, we consider the biterrorrate (BER) performance of our proposed MISO and MIMO transmit beamformer with perfect CSI available at the transmitter. For comparison purposes, we also implement several other designs. The "Con TxBm" denotes the conventional transmit beamformingf design without the uniform elemental power constraint. The "TxBm with ClIpping"!~) stands for the conventional design with peak pown~er clipping, which means that for every transmit antenna, if ws!1,i2 > We 2!1i Will be clipped by wtas = wtile7~l,/, ), i = 1,2,...,Nst. The "Heuristic SDR" refers to the Heuristic SDR solution described in Section 3.3.1. We denote "UEP TxBm" as the closedform MISO and the cyclic MIMO transmit beamformer designs under uniform elemental power constraint. Figure 32 shows the biterrorrate (BER) performance comparison of various transmit beamforming designs for both the (4, 1) MISO and (4, 2) MIMO systems. The "Con TxBm" achieves the best performance since it is not under the uniform elemental power constraint. Under the uniform elemental power constraint, the "UEP TxBm" schemes have much better performance than the "TxBm with Clipping." At BER = 103, for example, the improvement is about 1.5 dB for the (4, 2) MIMO system. In the MIMO system, we note that our "UEP TxBm" achieves almost the same performance as the "Heuristic SDR." Interestingly, if we increase both the transmit and receive antennas to 8, as shown in Figure 33, our "UEP TxBm" outperforms the "Heuristic SDR." The performance degradation of "Heuristic SDR" is caused by reducing the high rank optimal solution to (38) to a rankone solution heuristically. We note here that our "UEP TxBm" is also much simpler than the "Heuristic SDR" (see the discussions in Section 3.3). We examine next the effects of the two quantization methods (SQ and VQ) on the overall system performance. We use herein the suboptimal combination of {Bi)}l described in Section 3.4.1 for SQ due to its simplicity (although the optimal one can provide a better performance). We show in Figures 3437 the BER performance of various quantization schemes for our proposed and conventional transmit beamformer designs, with various numbers of feedback bits (B = 2, 4, 6, 8). We note that VQUEP outperforms the AVQ for all cases. When the number of feedback bits is small (e.g., B = 2, 4), VQUEP can provide a similar performance as that of CVQ, even though the latter is not under the uniform elemental power constraint! The VQUEP performance approaches that of the perfect channel feedback for "UEP TxBm" when the number of feedback bits becomes larger (e.g., B = 8). However, CVQ needs more bits to approach the performance of its perfect channel feedback counterpart. By using relatively large numbers of feedback bits (e.g., B = 6, 8), we can reduce the gap between the suboptimal SQ method and VQUEP, since we have already reduced the number of parameters to be quantized for the scalar method due to imposing the uniform elemental power constraint. Moreover, Figure 38 shows the BER performance of various (2, 1) MISO systems. In this case, we know that the "Alamouti Code" [23] has full rate and satisfies the uniform elemental power constraint. Compared to the "Alamouti Code," our proposed transmit beamformer design can achieve more than 2 dB SNR improvement using only a 2bit feedback, via either the suboptimal SQ or VQITEP. Our proposed transmit heamformer design with a 2bit feedback also performs similarly to its CVQ counterpart. Finally, we examine the accuracy of the approximate degradation D,(B) of the receive SNR given in (:338) for the MISO case. We carry out MonteCarlo simulations for a (4, 1) system and plot the numerically simulated degradation results in Figure :39. The training sequence size is set to NV = 2"7, and the channel variance is o, = 1. We observe that the approximate degradation given in (:338) is very close to the numerically simulated one for any feedback bit number (or rate) B. However, the highresolution approximation given in (:339) has accurate prediction only at high feedback bit rates. Note also that the SQ and VQITEP perform similarly when the feedback bit number is relatively large, which means that our approximate degradation expression of the receive SNR given in (:338), which is obtained for VQITEP, can also be used for SQ for large B. 3.7 Conclusion We have investigated M1131 transmit heamformer designs under the uniform elemental power constraint. The original problem is a difficulttosolve nonconvex optimization problem, which can he relaxed to an e Iilinsolve convex optimization problem via SDR. However, the rank reduction from an optimal SDR solution to a rankone transmit heamfomer may degrade the system performance. We have shown that a closedform expression for the optimal MISO transmit heamformer design exists. Then we have proposed a cyclic algorithm for the MIMO case which uses the closedform MISO solution iteratively. This cyclic algorithm has a very low computational complexity. The numerical examples have been used to demonstrate that our proposed transmit heamformer designs outperform the conventional counterpart with peak power clipping. They can have a better performance than the Heuristic SDR solution as well. Furthermore, we have considered finiterate feedback techniques for our proposed transmit heamformer designs. A scalar quantization method has been proposed and shown to be quite effective when the number of feedback bits is relatively large (e.g., B = 6, 8 for a (4,1) or (4,2) system). We have also proposed a vector quantization approach referred to as VQITEP. When the number of feedback bits is small, VQITEP can provide the same performance as CVQ even though the latter is not subject to the uniform elemental power constraint. Interestingly, for a (2,1) system, our finiterate feedback schemes can achieve more than 2 dB in SNR improvement compared to the "Alamouti Code" at the cost of requiring only a 2bit feedback. Finally, we have studied the average degradation of the receive SNR caused by VQITEP for the MISO case and obtained an approximate closedform expression. This approximation has been shown to be quite accurate, and can serve as an accurate guideline to determine the number of feedback bits needed in a practical system. We remark in passing that MIMO transmit heamformingf has exhibited great potential for reliable wireless communications and most likely will be adopted into the nextgeneration wireless local area network (WLAN) standards. Although our discussions here focus on the frequency flat Rayleigh fading channels, our MIMO transmit heamformer designs can he readily extended to the frequency selective fading channels and used in, for example, MIMOOFDM hased WLAN systems. n (4,1) Con TxBm  + (4,1) UEP TxBm + (4,1) TxBm with Clipping S1002 10 10 6 8 10 12 106 6 Figure 32. Performance comparison of various transmit beamformer designs with perfect CSI at the transmitter: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. 4 2 0 2 4 SNR (dB) 4 2 0 2 4 6 8 SNR (dB) 10. 102~ m104 a (8,8) Con TxBm + (8,8) UEP TxBm 10~  + (8,8) TxBm with Clipping E(8,8) Heuristic SDR . 106 1 0 8 6 4 2 0 2 SNR (dB) Figure 33. Performance comparison of various transmit beamformer designs for the (8,8) MIMO case. 10 m o, (4,1) 2bit SQ + (4,1) 2bit VQUEP k (4,1) 2bit AVQ 104 t (4,1) 2bit CVQ ll (4,1) UEP TxBm c (4,1) Con TxBm 10s 6 4 2 0 2 4 6 8 10 12 SNR (dB) (a) 100 10 102 o 104~ 0 (4,2) 2bit SQ + (4,2) 2bit VQUEP =k (4,2) 2bit AVQ s 1 (4,2) 2bit CVQ 10 5 i (4,2) UEP TxBm n (4,2) Con TxBm 106 6 4 2 0 2 4 6 8 SNR (dB) (b) Figure 34. Performance comparison of various transmit beamformer designs with 2bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. 10  !102 m I4 (4,1) 4bit SQ + (4,1) 4bit VQUEP + (4,1) 4bit AVQ 14 1 (4,1) 4bit CVQ ll (4,1) UEP TxBm c* (4,1) Con TxBm 10s 6 4 2 0 2 4 6 8 10 12 SNR (dB) (a) 100 10 12 10 104 _a (4,2) 4bit SQ +b (4,2) 4bit VQUEP. +I (4,2) 4bit AVQ 1 (4,2) 4bit CVQ 10 5 l (4,2) UEP TxBm ** (4,2) Con TxBm 106 6 4 2 0 2 4 6 8 SNR (dB) (b) Figure 35. Performance comparison of various transmit beamformer designs with 4bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. 10~  !102 4_ (4,1) 6bit SQ + (4,1) 6bit VQUEP + (4,1) 6bit AVQ 104 t (4,1) 6bit CVQ ll (4,1) UEP TxBm c (4,1) Con TxBm 10s 6 4 2 0 2 4 6 8 10 12 SNR (dB) (a) 100 10 102 104 e (4,2) 6bit SQ + (4,2) 6bit VQUEP s + (4,2) 6bit AVQ 5 c (4,2) 6bit CVQ 10 5 i (4,2) UEP TxBm a (4,2) Con TxBm 106 6 4 2 0 2 4 6 8 SNR (dB) (b) Figure 36. Performance comparison of various transmit beamformer designs with 6bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. !10  ~102 4_ (4,1) 8lbit SQ + (4,1) 8lbit VQUEP + (4,1) 8lbit AVQ 104 t (4,1) 8lbit CVQ ll (4,1) UEP TxBm c (4,1) Con TxBm 10s 6 4 2 0 2 4 6 8 10 12 SNR (dB) (a) 100 10 102 m1 4 0 (4,2) 8bit SQ 10 (4,2) 8lbit VQUEP + (4,2) 8lbit AVQ 1 (4,2) 8lbit CVQ 10~  ll (4,2) UEP TxBm a (4,2) Con TxBm 106 6 4 2 0 2 4 6 8 SNR (dB) (b) Figure 37. Performance comparison of various transmit beamformer designs with 8bit feedback: (a) the (4,1) MISO case, and (b) the (4,2) MIMO case. ~ (2,1) 2bit SQ + (2,1) 2bit VQUEP c (2,1) 2bit CVQ ll (2,1) UEP TxBm : ** (2,1) Con TxBm + (2,1) Alamouti Code I 102~ 10 3 104 4 2 0 2 4 6 8 10 12 SNR (dB) Figure 38. Performance comparison of various (2,1) MISO systems. 1 1.5 2 2.5 3 3.5 Average Degradation of Receive SNR Figure 39. Average degradation of the receive SNR for a (4,1) MISO system. CHAPTER 4 EFFICIENT CLOSEDLOOP SCHEMES FOR MIMOOFDM BASED WLANS 4.1 Introduction The singleinput singleoutput (SISO) orthogonal frequencydivision multiplexing (OFDM) systems for wireless local area networks (WLAN) defined by the IEEE 802.11a standard can support data rates up to 54 Mbps [72]. Improving the data rate to over 100 Mbps is a 1!! r ~ goal of the nextgeneration WLANs [73], [74]. The multiinput multioutput (1\![\!O) communication technology is widely regarded as a key to achieve such a high data rate. Assuming that the channel state information (CSI) is available at both the transmitter and the receiver, the MIMO channel can he decoupled, using singular value decomposition (SVD), into multiple orthogonal subchannels (or eigenmodes) on each subcarrier [75]. To maximize the channel throughout, power allocation and hit loading should be applied to the subchannels in both the spatial and frequency domains (see, e.g., [76] and the references therein). However, hit loading is often not adopted in practice, such as in the IEEE 802.11 standards, due to its complexity. If the same constellation is used across all the subchannels, the weaker eigfenmodes corresponding to the smaller singular values of the channel matrices tend to experience deeper fading [75], which degrades the overall system performance significantly. In [77], a power allocation method was proposed based on the minimum meansquared error (iljl~lmi) criterion for the MIMO systems. This method tends to put more power on the weaker subchannels, which may cause significant capacity loss. In this chapter we propose simple and efficient closedloop designs for MIMOOFDM hased WLANs. We focus on the IEEE 802.11a standard, although our schemes are also applicable to other standards including the U.S. standard IEEE 802.11g and the European standard HIPER LAN/2 [78]. Our schemes combine the recently proposed geometric mean decomposition (GMD) and uniform channel decomposition (UCD) transceiver designs [34], [35] with horizontal encoding and successive (noniterative) decoding. (An idea similar to GMD appeared in the independent work of [79].) GMD and UCD decompose each MIMO channel into multiple equal gain subchannels for each subcarrier, which allows our designs to obviate the need of any power allocations. The simulation results show that our closedloop schemes enjoy multidB improvement compared to the standard singular value decomposition (SVD) based schemes as well as the openloop VBLAST (vertical Bell Labs 1 Iwere 4 SpaceTime) based counterparts. In the explicit feedback mode, precoder feedback is required for the proposed schemes. We present a vector quantization algorithm for efficient precoder quantization. This quantization algorithm is inspired by an observation of the interesting link between a 2 x 2 unitary matrix and a 2D unit sphere. We show that the 2 x 2 unitary precoder matrix for each frequency subcarrier can be quantized by 6 bits with very small performance degradations. In the timedivision duplex (TDD) mode, where the channel reciprocity principle holds [74], our schemes do not require any precoder feedback. With a simple modification, our schemes can be made quite robust against the uplinkdownlink channel mismatches. The remainder of this chapter is organized as follows. Section 4.2 describes the 2 x 2 MIMO channel model with spatial correlations. Section 4.3 presents our closedloop MIMO WLAN system configuration, including the precoder and equalizer designs, and the successive soft decoding approach. In Section 4.4, we consider the explicit feedback mode and provide two quantization methods for precoder feedback, where a new vector quantization algorithm is proposed. In Section 4.5, we consider the TDD mode and show that the proposed schemes can be made quite robust against the uplinkdownlink channel mismatches. Numerical examples are given in Section 4.6 to demonstrate the effectiveness of our schemes. We conclude the chapter in Section 4.7. 4.2 Channel Model Consider a 2 x 2 MIMO channel with spatial correlations, where the channel can be modeled as [80] hii(t) hi12 2 h(t) = R)2 1 with Rt and R, quantifying the spatial correlations of the channel fading at the transmitter and the receiver, respectively, and L1 l=0 denoting the frequencyselective channel link between the jth transmit and ith receive antennas (with L being the channel length and T, the sampling period). The random variables { h } ~, I = 1,. ., L, are assumed to be independently distributed zeromean, circularly symmetric complex Gaussian variables. Applying Neipoint fast fourier transform (FFT) to h(t), we obtain the channel response in the frequency domain L1 H(u,) = ( Ne ,1< c (43) l=0 We denote Hk, ^H(f (k))=I H,,(f (k)) H12(f (k)) 1 the flat fading channel matrix at the kth data subcarrier (No, = 64 and NV = 48 for IEEE 802.11a), with f(k), 1 < k < NV, denoting the data subcarrier mapping function in [72]. 4.3 ClosedLoop MIMO WLAN System Design 4.3.1 System Description Our MIMOOFDM transmitter scheme is shown in Figure 41. We adopt the horizontal encoding method [81], where the two parallel branches perform encoding, bit interleaving, and data mapping separately. Let xik denote the ith encoded data symbol, i = 1, 2, on the kth subcarrier, and let xk 1k 2k] x~~. On each of the NV data subcarriers, the transmitter applies a 2 x 2 precoder matrix Pk, to obtain xk PkXk, 1 < k < NV. Denote xik the ith element of Xk. Then xik is the symbol to be transmitted in the ith branch at the kth subcarrier. Consequently each preceded branch is OFDM modulated using an Neipoint IFFT and is added with a cyclic prefix (CP) before transmission. The length of CP is assumed to be longer than the channel length L, and therefore the intersymbol interference (ISI) can be completely eliminated at the receiver side. In the explicit feedback mode, the precoders {Pk =1 are calculated and quantized at the receiver, and then fed back from the receiver to the transmitter. In the TDD mode, where the channel reciprocity principle holds, once the transmitter estimates the reverse channel, i.e., the one from the receiver to the transmitter, via training pilots, it can calculate the precoders Pk, k = 1,...,NV, to be used in the forward channel, i.e., from the transmitter to the receiver. Assuming accurate synchronization, frequency offset estimation and channel estimation, the receiver first removes the CP and applies an Neipoint FFT to each received branch as shown in Figure 42. Then the received signal vector at the kth data subcarrier is yk = HkPkXk Zk, k = 1, 2, ..., N, (45) where zk, ~ N(0, oj2I) denotes the circularly symmetric complex Gaussian noise. The key components of our closedloop designs are the precoder Pk, at the transmitter and the corresponding equalizers at the receiver, as we describe next. 4.3.2 Precoder and Equalizer Design We design the precoder and equalizer based on our GMD and UCD transceiver design schemes [34], [35]. Both schemes are based on the following theorem [34]. Theorem 4.3.1. Any rank K matrix: He C@xN with .:,li;larr UalueS XH,1 > H,2 *** AH,K > 0 can be decomposed into H = QRP*, (46) Figure 41. Transmitter design for MIMOOFDM based WLAN. Figure 42. Receiver design for MIMOOFDM based WLAN. where ()* denotes the conjugate transpose, R E RWKxK iS GR uppeT tn,:0ifluitlrr matrix: with equal diagonal elements ri = AH~ 1 < i < K, and Q E C~xK ,dp P E NxK GTC semiunitary matrices. Consider the channel model (45). In the explicit feedback mode, the GMD scheme [34] starts with the GMD matrix decomposition Hk, k k~PL at the receiver, to obtain Pk, Which is the unitary precoder to be fed back to the transmitter. Utilizing the precoder Pk, at the transmitter as in (45) leads to the following received data vector: yk = k~~x k+ Zk, k = 1, 2, ..., NV. (47) At the receiver, multiplying yk by Q* yields Yk k= k~~+z,(8 where Rk, = QLHkPk is a 2 x 2 upper triangular matrix with equal diagonal and Zk ~ NV(0, a 2I). The information symbols in xk can then be detected successively starting from the second element of Xk (See Section 4.3.3). The UCD scheme [35] is somewhat more complicated than GMD. Like GMD, the UCD scheme has two implementations forms, of which one can be regarded as a combination of a linear precoder with an MMSE VBLAST equalizer. Compared to GMD, which suffers from capacity loss at low to moderate SNR, UCD is strictly capacity lossless an d c an achi eve t he opt imal divers ity mult iplexi ng gai n t radeoff [ 21]. The det ails are omitted here due to limited space. Both GMD and UCD obviate the need of bit loading and power allocation at the transmitter and require only the feedback of the unitary precoders Pk, k = 1,...,NV. In the TDD mode, the forward channel is estimated at the transmitter and therefore the precoders Pk, can be calculated at the transmitter. 4.3.3 Successive Soft Decoding Note that Rk, is an upper triangular matrix. As shown in Figure 42, we adopt the schemes of deinterleaving, softdemapping and the lowcomplexity soft Viterbi decoder used in [73] for each branch separately. We first detect the data sequence of the lower branch to get the soft information. Assuming successful decoding of the data of the lower branch, we can cancel the interference due to the lower branch completely before decoding the upper branch, as is denoted by the feedback link at the lower part of Figure 42. The interference cancelation process of each subcarrier k using GMD is outlined as follows: 1) Initial Stage Calculate k=E [ 2k ki)22 2] a2(k 27 i2k = 2k (k)22, k~ = 1,... NV, where (Rk)ij, i.) = 1, 2, is the (i, j)th element of Rk, and .02k is the second entry of yk. Note that ak along with i2k proVides the soft information for the lower branch. We can decode the lower branch data sequence by using the soft Viterbi decoder. 2) Cancellation Stage Calculate k =, E [51lk (k)11 2] ~( k)~ ~21 and ilk ~ ~ ~ ~ j Ik k)2 k)1 k=1.., NV, where aSk along with f lk proVides the soft information for the upper branch. Here x12 is the reconstructed data symbol sequence obtained from the Viterbidecoder of the lower branch. Note that aSk k becauSe Rk, has equal diagonal. Given the soft information for the upper branch, we can also decode the upper branch data sequence by using the soft Viterbi decoder. For UCD, th ucesvesf decoding preur is,;,,,,,, similar., Because Ok k, the two branches have effectively the same output SNR. In contrast, the SVD hased or the conventional VBLAST based methods lead to two subchannels with unbalanced gains. For the systems with a fixed symbol constellation across all the subchannels, the weaker subchannel dominates the overall packeterrorrate (PER) performance, although iterative decoding between the two branches is helpful for reducing the PER of VBLAST [81]. 4.4 Precoder Quantization In the explicit feedback mode, the channel is estimated at the receiver. We compute the precoders Pk, k = 1,...,NV, at the receiver and feed them back to the transmitter. In the following, we present two quantization approaches to reduce the overhead of precoder feedback. 4.4.1 Scalar Quantization A simple scalar quantization scheme is as follows. Note that a 2 x 2 unitary precoder can be represented by cos 8 sin 8ej P(0, 4) = Oi Denote E(Ax, .) = cos 8,, sin Os, e ** (410 P(H, ~ Isin 8, e =2z cos 8,1 40 where Os,, =. xO the quantization levels of Os, and ~., respectively. After obtaining the precoder Pk, using GMD or UCD, we quantize Pk, to the " for each subcarrier k, we only need to feed the index (nl, n2) back to the transmitter, which requires log2 lV1V2) bits. To reduce the effect of quantization error and improve the robustness for GMD, instead of applying the original equalizer Q* at the receiver, we instead ulse Qg obtained. by the QRn decomposition: HkP 8ni _) k k, k =1,...,NV. (411) Note that P(0,,,' .) is known at the receiver. We also need to replace Rk, by Rk, in our interference cancelation stage. Clearly, when NI~ and N2~ arT TreSonably large, Rk~ is approximately equal to Rk, and the two diagonal elements of Rk~ are almost the same, i.e., the gains of the two branches remain almost the same. However, larger NI~ and N2~ also mean more feedback overhead. In practice, we need to chose NI~ and N2~ to achieve a reasonable tradeoff between feedback overhead and performance. Similarly, we can apply the MMSE VBLAST algorithm [82] to HkP 8ni _) oO obtain the equalizer when using UCD. 4.4.2 Vector Quantization Vector quantization can he adopted to further reduce the overhead of precoder feedback. We present a geometric approach to perform vector quantization. Suppose we quantize the precoder P(0, 4) to be P(0, ~), where (0, ~) correspond to an element in a codebook known to both the transmitter and receiver. Instead of transmitting the desired data vector P(0, ~)Xk at the transmitter, where Xk is the encoded data vector, we transmit P(0, ~)Xk. To optimize the quantization scheme, we minimize the following cost function i = E P(0, d)Xk P~g r8, IXk = : Ex [P(0, C) P (0] ) P (0, o) P (0, C)] X] , with respect to 8 and ~. This cost function measures the average distortion caused by the finite rate precoder quantization. Here the expectation is over Xk. After some straightforward algebra, we obtain =2I2 2 cos n cos B + sin sinl 0cos(O )] 2. (412) Because the value of E [ x, 2] does not affect our quantization problem, without loss of generality, let E [ 2] = 1. Then S= 2 2cos 0 os 0 sin 0 sinl i cos(d ) a 2 2(. (413) In the following, we give a geometric interpretation of (. We note that there is a onetoone and onto mapping from the unitary precoder set {P(0, 4) : 0 < 0 < xr, O < < 2x}) to the 2D unit sphere {ve R S : v = 1}. Any point on the 2D unit spherre with angles (0, ) lan he represented as v = Ilosi 0 in 0 osy ; in 0 sin 1 in the Cartesian coordinate, where the first element of v is the (1, 1)element of P(0, 4) and the second and third elements of v, respectively, are the real and imaginary parts of the (2, 1)element of P(0, ~). Each P(0, ~) corresponds to a point v on the 2D unit sphere. Similarly, any point on the 2D unit sphere with angles (0, 4) can be represented by the Cartesian coordinate v = cose sin, .l cos I sin 0 sn WeC sehat ( is just thei 1inner product between v and v. Define as the angle between v and v. Then ( = cos and d = (2 2 cos ~) = v v2 Based on this derivation, we conclude that a good codebook {Vs},), should be distributed on the unit sphere as uniform as possible. We use the following steps to determine the codebook. First, we generate a training set {va,a = 1,2,...,1V} via randomly picking 1V points on the 2D unit sphere, where 1V is a very large number. Next, starting with an initial codebook (obtained via the splitting method [64]), we iteratively update the codebook [64] until no further improvement on the minimum distance is observed based on the following criteria. 1) Nearest neighbor condition (NNC): for a given codebook {9 },)N1 assign a vector v, to the ith region Se= { e: , 942 V, V where Si, i = 1, 2, .. 4, is the partition set for the ith code vector. 2) Centroid condition (CC): for a given partition Si, the updated optimum code vectors {Vs},), satisfy vs= argr mmn E [vlL 42vL e 'S.L], i= 1, 2,...,411. (4 15) As shown in Appendix C, the solution to the above optimization problem is vi= ,i = 1 ,..., (416i) Iv iI wherevi= vntSs ] is the mean vector for the partition set S,,i = 1,2,...,1V. Hence, for each subcarrier k, we first map the precoder Pk, as a point v on the 2D unit sphere. According to the NNC criterion, we obtain the quantized vector v from the codebook with index i. The index i is fed back to the transmitter to reconstruct the precoder P(0, ~). In this case the overhead of feedback is log2(NV,) hits per subcarrier. 4.5 Robust Transceiver Design in the TDD Mode In the TDD mode, the channel reciprocity can he exploited to obviate the need of precoder feedback in high throughput MIMO WLAN system [74]. However, there is alr ima mismatch between the forward channel (from transmitter to receiver) and reverse channel (from receiver to transmitter) due to channel variations and/or amplifier mismatches, which poses 1!! I r~~ difficulties of utilizing the conventional closedloop schemes [83]. Our closedloop schemes can he modified to be robust against the mismatches and be backward compatible with the standard openloop VBLAST receiver [34]. Denote Hk, the forward channel assumed by the transmitter and Hk, the actual channel matrix at the kth data subcarrier. We may denote the channel mismatch as follows: Hk, = Hk, + cOE, 1 < k < N, (417) where E is a matrix whose elements are independently and identically distributed (i.i.d.) E[IHk,1 2 complexvalued Gaussian variables with zeromean and variance O.2 4 and a~ determines the level of channel mismatch. At the transmitter, the precoders Pk, k = 1,. ,N, are obtained based on Hk, k = 1,. ,N. The pilot (for channel estimation) and data sequences are both preceded using precoders Pk, k = 1,...,NV, before transmission, which leads to the following received signals instead of (47): Yk = HkPkXk Zk, k = 1, 2, ..., NV. (418) Assuming perfect channel estimation at the receiver, the estimated channel matrix on the k~th danta sulbcarrier is the "virtulal channel" HkPk. As inl Figulre 42, an equlalizer Qg is applied to the kth subcarrier to yield yk = QL(kk~lkl Zk TL), (419) where the equalizer Qg is obtained from the QR decomposition of HkPk, i.e., HkPk Qk k. Hence Yk R" k k Zk, and we can apply successive soft decoding as described in Section 4.3.3 to retrieve the transmitted data on the kth data subcarrier. Note that the channel gains of the two branches are usually unbalanced due to the mismatches between Hk, and Hk. However, for some small a~, the output SNRs of the two branches should be close, which results in only marginal performance loss, as shown with numerical examples in Section 4.6. Similarly, for UCD, the precoder Pk, is calculated according to the UCD procedure based on Hk, and the receiver involves an MMSE VBLAST equalizer. 4.6 Numerical Examples We present several numerical examples to demonstrate the superior performance of the proposed schemes. The system parameters used here are based on the IEEE 802.11a standard. For the two transmit and two receive antenna system, the 64QAM modulation and the channel coding rate of R = 3/4 are used. The total frequency bandwidth is 20MHz, which are divided into 64 subcarriers, including 48 data subcarriers. For each OFDM symbol with length 64 there is CP with length 16 which are discarded at the receiver to remove ISI Therefore the total data rate is 2 x log2 64 x x 48 x 20 x =4 108 Mbps. The channel between each transmit and receive antenna pair is generated according to the C'!s li It model [84] with 50 ns rootmeansquared (RMS) delay spread (here the sampling period is T, = 50 ns). We assume that the channels are perfectly estimated at the receiver. The data are formatted into packets consisting of 1000 information bytes. According to IEEE 802.11a, the goal is to achieve the packet error rate (PER) of 0.1. For the purpose of comparison, we also intplenient the following three standard schemes. The first is a simple SVD hased scheme. For this scheme, both the transmitter and receiver apply unitary rotations to diagonalize the channel matrix at each subcarrier, which yields 2 x 48 = 96 orthogonal data subchannels. No hit allocation is involved here, since otherwise 256QAM or larger constellations would be used, which would pose difficulties in the hardware intplenientations due to the phase noise issues, etc. The input power is uniformly allocated to all the 96 data subchannels. One encoder/decoder is sufficient in this case because the SVD completely eliminates the interference between subchannels and no successive decoding is needed. The second scheme is similar to the first, except that the power allocation (PA) algorithm of [77] is applied at each subcarrier. Because the two subchannels at each subcarrier is usually highly unbalanced, this power allocation algorithm tends to compensate the weaker one with more power. The third is an openloop MAISE VBLAST based scheme. Just like the proposed GMD and ITCD schemes, it applies two independent encoders and decoders for successive interference cancelation. Of course, unlike GMD and ITCD, the two data branches have usually unbalanced channel gains. Iniprovenient can he achieved via iterative decoding as described in the following. After decoding the lower branch, we can decode the upper branch with the influence front the lower branch canceled. Now given the decoded data front the upper branch, we can obtain improved decoding of the lower branch by removing the influence of the upper branch front the data. This procedure can he iterated. We also include the channel outage probability curve as a benchmark. C'I .Ill., I outage probability is defined as the probability that the instantaneous mutual information of the channel , I(SNR) = 20 x xx4 x4 log detI I2 + HH( 64 64 +16 2 is less than 108. The channel outage probability is the lower bound of the PER performance of any MIMO scheme. An informationtheoretically optimal scheme combined with a capaci' i 1!; Ob ving code should be able to achieve this curve. First, we consider channels without spatial correlation, i.e., R, = I2 and Rt = I2 (Cf. (41)). Figure 43 shows the PER performances of the proposed GMD/UCD schemes, the closedloop SVD with and without PA, and the openloop MMSE VBLAST [85] based scheme. We assume perfect precoder feedback. It can be seen from Figure 43 that the proposed closedloop designs have more than 4 dB SNR improvement over the MMSE VBLAST scheme at PER equal to 0.1, although one can have a 1 dB gain by applying iterative decoding. The SVD based method without PA has performance inferior to the openloop MMSE VBLAST based scheme. The scheme based on SVD with PA performs better, but there is still more than 3 dB loss compared to the GMD and UCD schemes. The dashed line represents the performance of the 802.11a system with data rate 54 Mbps. It is remarkable that compared with the SISO scheme, our simple closedloop 2 x 2 schemes can double the data rate and at the same time save 2.5 dB in total transmission power. Moreover, the PER curves of the GMD and UCD schemes have decreasing slopes much steeper than the other methods, which implies much improved diversity gain. There is still a gap of about 4.5 dB between the UCD scheme and the outage probability curve. Combined with a capacity achieving code, such as a Turbo code and a low density parity check code (LDPC) [86], the proposed schemes should close the gap further. Figure 44 shows a typical example of the output SNRs of the eigensubchannels ( + and 0) obtained by SVD at the 48 data subcarriers. We see that the weaker eigensubchannels has very low output SNR ( w,, less than 0 dB). These weak subchannels may cause too many detection errors for the error control code to handle. However, at each subcarrier, the GMD and UCD schemes decompose a MIMO channel into two identical subchannels. The output SNRs of the subchannels of GMD and UCD are also shown in Figure 44. We can see that the quality of the subchannels of GMD and UCD are much more stable across the subcarriers. This figure provides insight into the reason why GMD and UCD performs significantly better than the SVD based methods. We can also see that UCD outperforms GMD when the channel is close to singular, like the one at the 30th subcarrier. In the second example, we consider a spatially correlated channel with 1 0.7 1 0.3 R, = ,t Re 071 031 while all the other parameters remain the same as the first example. The results are given in Figulre 45. Compared with Figure 43, in Figulre 45 all the MIMOOFDM schemes suffer from performance degradations due to the spatial fading correlations. However, the relative advantage of the proposed closedloop schemes is even more prominent in this scenario. Specifically, the UCD scheme has a more than 4 dB gain over the SVD based schemes and approximately a 6 dB gain over the openloop MMSE VBLAST scheme at PER equal to 0.1. Indeed, we expect that the eigensubchannels obtained by SVD should have more disparate channel gains in the presence of fading correlations. Despite the fading correlations, the proposed GMD and UCD systems at the 108 Mbps data rate still provide better PER performance than the SISO system at half the data rate. We consider next the effect of quantized precoder on system performance. We use 8bit scalar quantization with NI~ = 24 and N2~ = 24 (cf. Section 4.4.1) and mbit vector quantization with NV, = 2m, m = 2, 4, 6, (cf. Section 4.4.2) to quantize the precoder Pk, Of each data subcarrier. Figures 46 and 47 show that the 6bit vector quantization performs equally well as the 8bit scalar quantization. By using the 6bit vector quantization, our quantized closedloop MIMO schemes suffer from less than 0.3 dB SNR loss compared to the perfect feedback case at PER=0.1. This small loss is negligible compared to the significant improvement of our proposed scheme over others. When more bits are used, we can further close the small gap. Finally, we consider the TDD mode. Figures 48 and 49 show that our closedloop schemes are quite robust against the mismatches between the channel matrices obtained at the transmitter and the receiver. Our closedloop schemes suffer from less than 0.2 dB loss at PER=0.1 even when the error parameter is as high as a~ = 0.1. 4.7 Conclusions We have presented simple and efficient closedloop designs for MIMOOFDM hased WLANs as a promising technology for the nextgeneration wireless LAN communications. By combining the recent GMD and UCD transceiver designs and the horizontal encoding architecture, we can achieve multidB improvement over the closedloop SVD hased schemes and the openloop MAISE VBLAST architecture. The advantage of our schemes is even more prominent when the fading channels are spatially correlated. We have also proposed an efficient algorithm for the quantization of 2 x 2 unitary precoders. Using only a 6bit vector quantization at each data subcarrier, the system can achieve performance very close to the perfect precoder feedback, which represents a very moderate feedback overhead in the explicit feedback mode. In the TDD mode, when the channel reciprocity mechanism is available, we can modify our closedloop designs to be robust against the mismatches between the forward channel and reverse channel. The extensive numerical experiments validate the superior performance of the proposed schemes. Finally, we remark that, although our discussions focus on the 2 x 2 system, our schemes can he readily extended to the case of more transmit and more receive antennas. S2 W10 .  Outage Probability I:: ** MMNSE. a MMNSE Ite=3 10 SISO (54Mbps) 4 SVD + SVD with PA I e UCD 104 14 16 18 20 22 24 26 28 SNR Figure 43. Performance comparison of MIMO WLAN (108 Mbps) schemes for uncorrelated channels in the absence of quantization errors. 30 25 20 *C1 15 0? 5 10 O 20 30 Index of Data Subcarrier Figure 44. Output SNRs of the subchannels obtained via GMD, UCD, and SVD, with input SNR= 22 dB.  Outage Probability a MMNSE , SISO (54Mlbps) 4 SVD + SVD with PA GMID e UCD   0 15 20 102~ 1 \ a . I . i I 1. I 1 0 4 1( SNR Figure 45. Performance comparison of MIMO WLAN (108 Mbps) channels in the absence of quantization errors. schemes for correlated 100 ~ o LU S12 10 20 21 22 23 24 SNR 25 26 27 28 Figure 46. Performance comparison of the proposed closedloop schemes for uncorrelated channels with scalar or vector quantization for GMD. 1  a MMSE  <1 UCD bits VQ > UCD 4bits VQ : n UCD 6bits VQ 4 UCD 8bits SQ e UCD Perfect Feedback 10 20 21 22 23 24 25 26 27 28 SNR Figure 47. Performance comparison of the proposed closedloop schemes for uncorrelated channels with scalar or vector quantization for UCD. 100 & 10 S12 10 20 21 22 23 24 25 26 SNR Figure 48. Performance comparison of the proposed closedloop schemes for uncorrelated channels under channel mismatches in the TDD mode for GMD. g, 1 0 1': .   10 20 21 22 23 24 25 26 SNR Figure 49. Performance comparison of the proposed closedloop schemes for uncorrelated channels under channel mismatches in the TDD mode for UCD. CHAPTER 5 CONCLUSIONS AND FITTIRE WORK( My research has created and tested several spacetinle processing algorithms in niultiantenna systems for wireless coninunications. The contribution of my research has been threefold: First, we have presented several adaptive heanifornlingf methods based on the nmaxintizing the SINR, for data detection in broadband coninunications in the presence of unknown cochannel interference (CCI). CCI has been a bottleneck which severely limits the capacity of broadband wireless systems. Our AD algorithm first obtains a spacetinle heanifornier to suppress the CCI by using the received data, and then applies the Viterbi algorithm for symbol detection or possibly uses only single symbol detection. R AD considers combating the detrimental effect of the practical imperfect channel estimation problem by reestiniating the "true" channel. The iterative versions of AD and R AD have also been proposed to further improve the detection performance. We have shown in the numerical examples that all our proposed methods have a significant performance intprovenient over the conventional MLSE methods. Second, we have studied a novel MIMO transmit heanifornxingf under uniform elemental power constraint. This scheme takes into account the practical intplenientation constraint of uniform elemental amplifier at each transmit antenna. The original design is a nonconvex optimization problem, and it is usually difficult to find the optimal solution. We have proposed a computationally attractive cyclic algorithm for the MIMO transmit heanifornier. Furthermore, we have investigated the finiterate feedback techniques for our proposed design. A simple scalar quantization method and a vector quantization method (VQITEP) have been presented, which have been shown to be quite effective. We have analyzed the average degradation of the receive SNR caused by VQITEP for the MISO case. The soobtained closedfornt expression can serve as an accurate performance prediction in a practical system. Third, we have proposed simple and efficient closedloop designs for MIMOOFDM hased WLANs. The SISO OFDM systems for WLANs defined by the IEEE 802.11a standard can support data rates up to 54 Mbps. By combining the recently introduced 2 x 2 GMD and UCD transceiver designs and the horizontal encoding architecture, we have not only doubled the date rates, but also achieved multidB improvement over the SISO counterpart. The performance improvement of our schemes is even more significant over the closedloop SVD hased schemes and the openloop MAISE VBLAST architecture at the same data rate. For the finiterate feedback, we have proposed an efficient vector quantization method for the 2 x 2 precoders, with a good geometric explanation. In the TDD mode, when the channel reciprocity is assumed, our closedloop designs have been shown to be very robust against the mismatches between the forward channel and reverse channel. Multiantenna systems with channel state information (CSI) at the transmitter is a hot research area and many open problems still exist. We discuss in the following some possible future directions. Efficient FiniteRate Feedback Schemes for Transmit Precoding We have studied in OsI Ilpter 4 the finiterate feedback for 2 x 2 MIMO transceiver design systems, where the 2 x 2 precoder can he expressed as a function of 2 parameters and mapped as a unique point on the surface of a unit 2D sphere. When more transmit antennas are deploi II more parameters are needed to characterize the precoder. One interesting problem is to construct an efficient mapping between the precoder and the parameters needed, for example, according to the importance of each parameter. For a MIMOOFDM system as discussed in OsI Ilpter 4, the cost of feeding back the required hits increases with the number of subcarriers. How to compress the CSI efficiently for feedback purposes by exploiting the relationships among the subcarriers could be another topic. Motivated by the practical power concern in C'!s Ilter 3 for MIMO transmit heamforming, we can also consider the AllMO preceding (or MIMO transceiver design) under the uniform elemental power constraint. This could be a more challenging problem since multiple weights are coupled together for each transmit antenna. Designing efficient finiterate feedback schemes for this design is important as well. Partial Transmit CSI for MultiUser Communications Exploiting the CSI at the transmitter, M1131 transmit heanifornxingf and M1131 preceding offers many advantages: additional array gain [28], [29], diversity and multiplexing tradeoff [:33], [:34], [:35], etc. When the full CSI is not available at the transmitter, finiterate feedback techniques can still provide a good performance for those schemes developed for the singleuser case, as shown in ChI Ilpters :3 and 4. In niultiuser coninunications, exploiting only partial CSI at the transmitter is an important open problem. The previous work has established the capacity regions for the broadcast channels and the Gaussian multiple access [12], based on the availability of full CSI at both the transmitter and the receiver. For example, dirty paper coding [87] is a capacityoptinmal scheme for the broadcast channel. However, the capacity of broadcast channel or the performance of intplenientable schemes (e.g., dirty paper [87]) for partial CSI remains unknown. Furthermore, additional research is needed to determine the performance of MIMO transmit heanifornxingf and M1131 preceding in the niultiuser case when coupled with nmultiuser schemes (e.g., dirty paper scheme for broadcast channel). APPENDIX A PROOFS FOR CHAPTER 2 We prove below that (234) is equivalent to (235). Fix ho in (234) and therefore consider the problem: max &2, s R.I &2 HoH > 0. (A1) Next note that I &11/2HoH~ R/ O 0 1 > b Amax,[Ri1/2 0 1/2l" o (A2) Xmin [(H R1Ho)1 It follows that the solution to (A1) is given by Si~ = Xmin 0~iH)1], (A3) and the proof of (234) is concluded. APPENDIX B PROOFS FOR CHAPTER 3 We prove that the average degradation D,(B) of the receive SNR in (338) is a monotonically decreasing function of the nonnegative real number B. We let b= 2B + (1 a)N l. Then the first derivative of D,(B) with respect to B is (41)In22 b (1aL)"t a (4 " 1)2bt1 ( In 2) e" c: 2"(1 )"L br1 bt1 2fi(1 a)" ] , D',(B) (B1) where c n (1V Note that 1) In 2 a~ is a constant. b =(1 a [ +2R( (1 a) l)"t Using the Taylor series expansion, we can expand b~t as ba1 = (1 ) [2 [ (1 )ih l R'(In n= o (B2) (B3) ba1 = 2N [28 (1 )NI" n= o (B4) (B5) whr f()(1 a = 0, wher f(" (lN1)" )(n 1 N ) n>1. Nt1 Nt1 Since I~()ll> lf"1!~l, w obtain the inequalities as follows: S(1 a) 1 + 2B (1 a)C' ) 2" <(1 2 1t 2B" 1 (1a 2B > (1 2 Nt 1 a)N l a)N l For the 2B < (1 a)N l case, we have (B6) For the 2B > (1 a)N case, we have (B7) Summarizing the above inequalities, we get D',(B) < 0. Thus, the average degradation of the receive SNR D,(B) is a monotonically decreasing function of the nonnegative real number B. D ',B ) c (1 a 1 1 a) (1+ 2)N "1 a "1 [I1 1 I B1aN c 1 %v 1 1 2 B (1 a)(N"2) < 0. D', (B) < c 2 "t 2"1) ] (1 +"( a) c2(1 a)t c B < 2 t [1 +(1V 1)(1 1) 1 1] = 0. Iv1 in 4 [ ; min  v,, 2 s.t. 942 1 The Lagrangian corresponding to the constrained optimization problem is (C1) 3 vu6S, l=1 l=1 (C2) From the first derivative conditions: d~(v= 0, S= 1, 2, 3, Cv6S, ~Unl , 131 v6IUj I =1, 2, 3. Ev i, 1,I= 1, 2, 3. 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